ĐỀ CƯƠNG HỌC TẬP LỚP 11
SỞ GD & ĐT TP. HỒ CHÍ MINH
TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ
Mơn: Tốn, Năm học: 2019 – 2020
§ 3. NHỊ THỨC NEWTON
1. Nhò thức Newton: Cho a, b là các số thực và n . Ta có:
(a b)n
n
C
k 0
(x 2)4
4
C
k 0
k
4
k
n
.a n k .b k C n0a n C n1a n 1b C n2a n2b 2 C nn 1ab n 1 C nnb n
.x 4k .2k C 40 .x 4 .20 C 41.x 3 .21 C 42 .x 2 .22 C 43 .x 1.23 C 44 .x 0 .24
x 4 8x 3 24x 2 32x 16.
(x 2y)5 ........................................................................................................................................
..................................................................................................................................................................
6
1
x ........................................................................................................................................
x
..................................................................................................................................................................
6
1
2x ......................................................................................................................................
x
..................................................................................................................................................................
2. Nhận xét:
Trong khai triển (a b)n có n 1 số hạng và các hệ số của các cặp số hạng cách đều số
hạng đầu và số hạng cuối thì bằng nhau: C nk C nnk .
Số hạng tổng qt dạng: Tn 1 C nk .a n k .b k và số hạng thứ N thì k N 1.
Trong khai triển (a b)n thì dấu đan nhau, nghĩa là , rồi , rồi , ….…
Số mũ của a giảm dần, số mũ của b tăng dần nhưng tổng số mũ a và b bằng n.
3. Tam giác Pascal:
Các hệ số của khai triển: (a b)0, (a b)1, (a b)2, ..., (a b )n có thể xếp thành một tam
giác gọi là tam giác PASCAL.
n 0:
n 1:
1
1 1
n 2:
1 2
1
n 3:
1
3
n 4:
n 5:
1 4 6 4 1
1 5 10 10 5
n 6:
1
6
15 20 15 6
n 7:
1
7
21 35 35 21 7
3
Hằng đẳng thức
PASCAL
C nk11 C nk1
1
1
C nk
1
1
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Trang - 127 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
Dng toỏn 1. Tỡm h s hoc s hng trong khai trin nh thc Newton
n
k
n
Cn nh: Tk 1 C a
n k
.b
n
k
v x .x
m
x
m n
1. Tỡm s hng khụng cha x (c lp vi x )
1
trong khai trin x , x 0.
x
1
trong khai trin x 3 2 , x 0.
x
1
v n 5.
x2
..............................................................................
k
1
S hng tng quỏt: Tk 1 C 5k .(x 3 )5k . 2
x
..............................................................................
..............................................................................
..............................................................................
k
(1)
C 5k .(1)k .x 155k .
2k
x
S hng khụng cha x 15 5k 0 k 3.
C 5k .x 153k .
2. Tỡm s hng khụng cha x (c lp vi x )
12
5
Ta cú: a x 3, b
x
xn
xn
, m x n m , (x .y )n x n .y n , n
y
x
y
Vy s hng cn tỡm l C 53 .(1)3 10.
3. Tỡm s hng khụng cha x (c lp vi x )
10
1
trong khai trin x 4 , x 0.
x
..............................................................................
..............................................................................
S: 924. ..............................................................
4. Tỡm s hng khụng cha x (c lp vi x )
12
x
3
trong khai trin , x 0.
3 x
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S: 45. .................................................................
S: 924. ..............................................................
5. Tỡm s hng khụng cha x (c lp vi x )
6. Tỡm s hng khụng cha x (c lp vi x )
6
1
trong khai trin 2x 2 , x 0.
x
10
1
trong khai trin 2x , x 0.
x
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S: 240. ...............................................................
S: 8064. ........................................................
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Trang - 128 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
7. Tỡm h s ca s hng cha x 16 trong khai 8. Tỡm h s ca s hng cha x 6 trong khai
trin nh thc (x 2 2x )10 .
trin nh thc (1 3x )11.
Ta cú: a x 2, b 2x, n 10.
..............................................................................
k
S hng TQ: Tk 1 C 10
.(x 2 )10k .(2x )k
..............................................................................
C 10k .x 2(10k ).(2)k .x k
..............................................................................
C 10k .(2)k .x 202k .x k
..............................................................................
C 10k .(2)k x 20k .
..............................................................................
Vỡ cú s hng x 16 20 k 16 k 4.
..............................................................................
H s cn tỡm l C 104 .(2)4 3360.
S: 336798. .......................................................
9. Tỡm h s ca s hng cha x 15 trong khai 10. Tỡm h s ca s hng cha x 4 trong khai
trin nh thc (3x x 2 )12 .
trin nh thc (x 3)9 .
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S: 4330260. ...................................................
S: 30618. ......................................................
11. Tỡm h s ca s hng cha x 12y 13 trong 12. Tỡm h s ca s hng cha x 8y 9 trong
khai trin nh thc (x y )25 .
khai trin nh thc (2x 3y )17 .
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S: 5200300. ......................................................
S: C 179 .2179.39 122494394880. ...................
13. Tỡm h s ca s hng cha x 6y 7 trong 14. Tỡm h s ca s hng cha x 25y 10 trong
khai trin nh thc (2x y )13 .
khai trin nh thc (x 3 xy )15 .
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S: 109824. ........................................................
S: 3003. ............................................................
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Trang - 129 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
15.
Tỡm h s ca s hng cha x 4 trong 16.
khai trin nh thc (1 x 3x 2 )10.
Ta cú: (1 x 3x 2 )10 [1 (x 3x 2 )]10
(a 1, b x 3x 2, n 10)
10
C 10k .110k.(x 3x 2 )k
k 0
10
C
k 0
k
10
(x 3x 2 )k
2
(a x, b 3x , n k )
10
k
C .C
k 0
10
k
10
p 0
p
k
.x k p .(3x 2 )p
k
C 10k .C kp .x k p .3p.x 2 p
k 0 p 0
10
k
C
k 0 p 0
k
10
.C kp .3p x k p .
Mụn: Toỏn, Nm hc: 2019 2020
Tỡm h s ca s hng cha x 17 trong
khai trin nh thc (1 x 2x 2 )10 .
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4
Vỡ cú s hng x k p 4 vi iu kin
0 p k 10, (p, k ) nờn cú bng:
..............................................................................
p
0 1 2
3
k 4 p 4 3 2 1 (L)
..............................................................................
Do ú h s ca s hng cha x 4 l:
..............................................................................
C 104 C 40 .30 C 103 C 31 .31 C 102 C 22 .32 1695.
S: 38400. .........................................................
17.
..............................................................................
Tỡm h s ca s hng cha x 8 trong 18.
khai trin nh thc (1 x 2 x 3 )8 .
Tỡm h s ca s hng cha x 3 trong
khai trin nh thc (x 2 x 1)5 .
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S: 238. ...............................................................
S: 5. ..................................................................
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Trang - 130 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
19.
Tỡm h s ca s hng cha x 10 trong 20.
khai trin P(x ) (1 x x 2 x 3 )5 .
Mụn: Toỏn, Nm hc: 2019 2020
Tỡm h s ca s hng cha x 5 trong
khai trin P(x ) (1 x x 2 x 3 )10 .
Ta cú: P(x ) [(1 x ) x 2 (1 x )]5
..............................................................................
[(1 x )(1 x 2 )]5
..............................................................................
(1 x )5 .(1 x 2 )5
..............................................................................
5
5
..............................................................................
k 0
p 0
..............................................................................
.x k 2 p .
..............................................................................
C 5k .15k.x k .C 5p .15p.x 2p
5
5
C C
k
5
k 0 p 0
p
5
..............................................................................
Vỡ cú s hng x
10
nờn k 2p 10 v cú bng
p
2
3
4
5
6
k 10 2p
6
4
2
0 2
(0 p; k 5) (L) (N) (N) (N) (L)
Do ú h s ca s hng cha x 10 l
4
5
3
5
2
5
4
5
0
5
5
5
C C C C C C 101.
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S: 1902. ............................................................
21. Xột P(x ) x (1 2x )5 x 2 (1 3x )10 . Tỡm 22. Xột P(x ) x (2x 1)6 (3x 1)8 . Tỡm h
h s x 5 trong khai trin P (x ).
5
10
..............................................................................
k 0
p 0
..............................................................................
Ta cú P (x ) x C 5k (2x )k x 2 C 10p (3x )p
.x .(2)k .x k C 10p .x 2 .3p.x p
10
..............................................................................
p 0
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S: 3320. ............................................................
S: 13368. ......................................................
5
s x 5 trong khai trin P (x ).
C
k 0
k
5
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Trang - 131 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
23.
Tỡm h s ca x 6 trong khai trin biu 24.
4
4
4x 2 4x 1
Ta cú: P (x ) (2x 1)
4
Tỡm h s ca s hng cha x 10 trong
2
1
thc P (x ) (2x 1) x 2 x .
4
x 2
khai trin P (x ) x 1 (x 2)15 .
4
6
6
Mụn: Toỏn, Nm hc: 2019 2020
..............................................................................
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4
(2x 1)2
(2x 1)6 .
4
8
(2x 1)
(2x 1)6 .
44
1
(2x 1)14 .
256
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ỏp s:
25.
3003
. .....................................................
4
Tỡm h s ca x 5 trong khai trin sau:
S: a10
26.
1 9 9
.C .2 2956096. .....................
16 19
Tỡm h s ca x 5 trong khai trin sau:
(2x 1)4 (2x 1)5 (2x 1)6 (2x 1)7 .
(x 1)6 (x 1)7 (x 1)8 (x 1)12
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ỏp s: 896. ........................................................
S: 1715. ...........................................................
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Trang - 132 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
27. Gi s cú khai trin a thc:
Mụn: Toỏn, Nm hc: 2019 2020
28. Gi s cú khai trin a thc:
(1 2x )n a 0 a1x a 2x 2 a n x n .
(1 4x )n a 0 a1x a 2x 2 a n x n .
Tỡm a 5 , bit rng a 0 a1 a2 71.
Tỡm a5 bit a 0 a1 a 2 1197.
Cú (1 2x )n
n
C
k 0
k
n
(2x )k
n
C
k 0
k
n
(2)k x k
..............................................................................
k
n
k
Dng tng quỏt ca h s l ak C (2) .
1
n
..............................................................................
..............................................................................
0
Vi k 0 a 0 C (2) 1.
..............................................................................
1
n
Vi k 1 a1 2C 2n.
..............................................................................
2
n
Vi k 2 a 2 4C .
..............................................................................
Theo bi ta cú: a 0 a1 a 2 71
..............................................................................
2
n
1 2n 4C 71 .................................
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ỏp s: a 5 C 75 (2)5 672. .......................
S: a5 1317888. .........................................
29.
Gi s cú khai trin a thc:
n
30.
Gi s cú khai trin a thc:
(1 2x ) a 0 a1x a 2x an x .
(1 2x )n ao a1x a 2x 2 a n x n .
Tỡm n bit a 0 8a1 2a 2 1.
Bit a 3 2014a 2 . Tỡm n.
2
n
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S: n 5. ..........................................................
S: n 6044. ....................................................
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Trang - 133 -
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020
BI TP V NH 1
Cõu 1. Cú bao nhiờu s hng trong khai trin nh thc (2x 3)2018 .
A. 2017.
B. 2018.
..............................................................................................
C. 2019.
D. 2020.
..............................................................................................
Cõu 2. Trong khai trin (a b)n , s hng tng quỏt ca khai trin l
A. C nka n kb k .
B. C nk 1a n 1b n k 1 .
..............................................................................................
C. C nka n kb n k .
D. C nk 1a n k 1b k 1 .
..............................................................................................
Cõu 3. Tỡm s hng cha x 3y 3 trong khai trin (x 2y )6 thnh a thc.
A. 120x 3y 3 .
B. 160x 3y 3 .
..............................................................................................
C. 20x 3y 3 .
D. 8x 3y 3 .
..............................................................................................
Cõu 4. Tỡm h s ca s hng cha x 3 trong khai trin nh thc Niutn (2x 1)6 .
A. 160.
B. 960.
..............................................................................................
C. 960.
D. 160.
..............................................................................................
Cõu 5. Gi s cú khai trin (1 2x )7 a 0 a1x a 2x 2 a 7 x 7 . Tỡm a 5 .
A. 672x 5 .
B. 672.
..............................................................................................
C. 672x 5 .
D. 672.
..............................................................................................
Cõu 6. Tỡm h s ca x 6 trong khai trin thnh a thc ca (2 3x )10.
A. C 106 .26.(3)4 .
B. C 106 .24.(3)6 .
..............................................................................................
C. C 104 .26.(3)4 . D. C 106 .24.36.
..............................................................................................
10
1
Cõu 7. S hng khụng cha x trong khai trin P (x ) 2x l s hng th
x
A. 6.
B. 7.
..............................................................................................
C. 8.
D. 9.
..............................................................................................
12
3 x
Cõu 8. H s ca s hng cha x trong khai trin nh thc (vi x 0 ) l
x 3
6
220
729
220 6
x .
C.
729
A.
220 6
x .
729
220
D.
729
B.
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ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
6
1
Cõu 9. S hng khụng cha x trong khai trin 2x 2 l
x
A. 60.
B. 120.
..............................................................................................
C. 480.
D. 240.
..............................................................................................
9
1
Cõu 10. H s ca s hng cha x trong khai trin x 3 , (vi x 0 ) bng
x
3
A. 36.
B. 84.
C. 126.
D. 54.
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Cõu 11. S hng cha x 4 trong khai trin (2 x )7 thnh a thc l
A. 8C 74 .
B. C 74 .
..............................................................................................
C. 8C 74x 4 .
D. C 74x 4 .
..............................................................................................
45
1
Cõu 12. S hng khụng cha x trong khai trin x 2 l
x
5
A. C 45
.
5
B. C 45
.
15
C. C 45
.
15
D. C 45
.
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Cõu 13. Trong khai trin ca (1 3x )9 s hng th 3 theo s m tng dn ca x l
A. 180x 2 .
B. 120x 2 .
..............................................................................................
C. 324x 2 .
D. 4x 2 .
..............................................................................................
21
2
Cõu 14. Tỡm s hng khụng cha x trong khai trin nh thc Newton x 2
x
7
A. 27C 21
.
B. 28C 218 .
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C. 28C 218 .
7
D. 27C 21
.
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12
1
Cõu 15. Cho x l s thc dng, khai trin nh thc x 2 ta cú h s ca s hng cha
x
x m bng 495. Tp hp giỏ tr ca m l
A. {4; 8}.
B. {0}.
..............................................................................................
C. {0;12}.
D. {8}.
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Trang - 135 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
n
1
Cõu 16. Bit h s ca s hng cha x trong khai trin 3x 2 l 34C n5 . Khi ú giỏ tr
x
ca n bng
3
A. 15.
B. 9.
C. 16.
D. 12.
Cõu 17.
B. 70.
C. 56.
D. 56.
B. 1272.
C. 1752.
D. 1272.
B. 263.
C. 632.
D. 956.
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H s ca x 4 trong khai trin a thc f (x ) x (1 x )5 x 2 (1 2x )10 bng
A. 965.
Cõu 20.
..............................................................................................
Tỡm h s x 5 trong khai trin x (2x 1)6 (x 3)8 .
A. 1752.
Cõu 19.
..............................................................................................
Tỡm h s ca s hng cha x 6 trong khai trin x 3 (1 x )8 .
A. 28.
Cõu 18.
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Gi s (1 x x 2 )n a 0 a1x a 2x 2 a 2n x 2n . t S a 0 a2 a 2n ,
khi ú S bng
A.
3n 1
2
3n
2
..............................................................................................
D. 2n 1.
..............................................................................................
B.
3n 1
C.
2
..............................................................................................
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P N BI TP V NH 1
1.C
2.A
3.B
4.D
5.B
6.B
7.A
8.A
9.A
10.B
11.C
12.C
13.C
14.D
15.C
16.B
17.C
18.D
19.A
20.A
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Trang - 136 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
BI TP V NH 2
Cõu 1. S hng tng quỏt trong khai trin ca (1 2x )12 l
k
A. (1)k C 12
2x k .
k k k
B. C 12
2 x .
..............................................................................................
C. (1)k C 12k 2k x k . D. C 12k 2k x 12k .
..............................................................................................
Cõu 2. H s ca x 5 trong khai trin (1 x )12 l
A. 820.
B. 210.
..............................................................................................
C. 792.
D. 220.
..............................................................................................
Cõu 3. H s ca s hng cha x 3 trong khai trin (1 x )10 l
A. 30.
B. 120.
..............................................................................................
C. 120.
D. 30.
..............................................................................................
Cõu 4. H s ca x 10 trong biu thc P (2x 3x 2 )5 bng
A. 357.
B. 243.
..............................................................................................
C. 628.
D. 243.
..............................................................................................
Cõu 5. Trong khai trin biu thc (x y )21, h s ca s hng cha x 13y 8 l
A. 116280.
B. 293930.
..............................................................................................
C. 203490.
D. 1287.
..............................................................................................
Cõu 6. Trong khai trin (a 2b)8 , h s ca s hng cha a 4 .b 4 l
A. 560.
B. 70.
..............................................................................................
C. 1120.
D. 140.
..............................................................................................
Cõu 7. Tỡm h s ca x 6 trong khai trin thnh a thc ca (2 3x )10 .
A. C 106 .26.(3)4 .
B. C 106 .24.(3)6 .
..............................................................................................
C. C 104 .26.(3)4 D. C 106 .24.36.
..............................................................................................
10
1
Cõu 8. H s ca x trong khai trin x 3 bng
x
6
A. 792
B. 210
..............................................................................................
C. 165
D. 252
..............................................................................................
7
2
Cõu 9. Tỡm h s ca s hng cha x trong khai trin x 2 .
x
5
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Trang - 137 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
A. 84.
B. 672.
..............................................................................................
C. 560.
D. 280.
..............................................................................................
6
1
Cõu 10. Tỡm s hng khụng cha x trong khai trin 2x 2 .
x
A. 15.
B. 240.
..............................................................................................
C. 240.
D. 15.
..............................................................................................
5
Cõu 11. Tỡm h s ca x
10
2
trong khai trin biu thc 3x 3 2 .
x
A. 240.
B. 810.
..............................................................................................
C. 810.
D. 240.
..............................................................................................
7
x 2 1
Cõu 12. Tỡm s hng cha x trong khai trin .
2
x
5
35 5
x .
16
16
C. x 5 .
35
A.
35 5
x .
16
16 5
D.
x .
35
B.
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 13. Xột khai trin: (5x 1)2017 a2017x 2017 a 2016x 2016 a1x a 0 . Giỏ tr a 2000 bng
17
A. C 2017
.517.
17
B. C 2017
.517.
..............................................................................................
17
C. C 2017
.52000.
17
D. C 2017
.52000.
..............................................................................................
7
1
Cõu 14. H s ca x trong khai trin ca x 2 (2x 1)2 bng
x
2
A. 4.
B. 40.
..............................................................................................
C. 35.
D. 39.
..............................................................................................
Cõu 15. Tỡm h s ca x 5 trong khai trin P (x ) (x 1)6 (x 1)7 (x 1)12 .
A. 1715.
B. 1711.
..............................................................................................
C. 1287.
D. 1716.
..............................................................................................
Cõu 16.
Tỡm h s ca s hng cha x 9 trong khai trin nh thc Newton (1 2x )(3 x )11 .
A. 4620.
B. 1380.
..............................................................................................
C. 9405.
D. 2890.
..............................................................................................
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Trang - 138 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Cõu 17.
Mụn: Toỏn, Nm hc: 2019 2020
Tỡm h s ca x 5 trong khai trin P (x ) x (1 2x )5 x 2 (1 3x )10 .
A. 3240.
B. 3320.
C. 80.
D. 259200.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 18.
Cho khai trin (1 2x )9 a 0 a1x a 2x 2 a 9 x 9 . Tng a 0 a1 a 2 bng
A. 127.
B. 46.
C. 2816.
D. 163.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 19.
Tỡm h s ca x 7 trong khai trin f (x ) (1 3x 2x 3 )10 thnh a thc.
A. 204120.
B. 262440.
C. 4320.
D. 62640.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 20.
Tỡm h s ca s hng cha x 5 trong khai trin (1 x x 2 x 3 )10 .
A. 582.
B. 1902.
C. 7752.
D. 252.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
P N BI TP V NH 2
1.C
2.C
3.B
4.D
5.C
6.C
7.B
8.B
9.D
10.B
11.C
12.C
13.C
14.D
15.A
16.C
17.B
18.A
19.D
20.B
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Trang - 139 -
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020
Dng toỏn 2. Chng minh hoc tớnh tng
n
(a b )n C nk .a n k .b k C n0a n C n1a n 1b C n2a n 2b 2 C nn 1ab n 1 C nnb n
k 0
S m ca a gim dn, s m ca b tng dn nhng tng s m a v b bng n.
Trong khai trin (a b)n thỡ du an nhau, ngha l , ri , ri , .
1
31. Chng minh: 316C 160 315C 16
314C 162 3C 1615 C 1616 216.
Suy lun:
S m ca s 3 gim dn Chn a 3.
Khụng cú s m ca s no tng Chn b 1 (vỡ 10 11 12 116 1).
Du an nhau (cng ri tr, cng tr..) nờn chn khai trin (a b)n (3 1)n .
Vỡ t hp dng: C 160 , C 161 , C 162 , ...C 1616 nờn chn n 16.
Li gii tham kho
Xột (3 1)16
16
C
k 0
16
16
k
16
.316k .(1)k 316C 160 315C 161 314C 162 3C 1615 C 1616
0
16
1
2 3 C 315C 16
314C 162 3C 1615 C 1616 (pcm).
32. Tớnh tng S C 50 2C 51 22C 52 25C 55 .
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S 35. .....................................................................................................................................
33. Tớnh tng S 4 0C 80 41C 81 42C 82 4 8C 88 .
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S 58. .....................................................................................................................................
34. Tỡm n tha món 3n C n0 3n 1C n1 3n 2C n2 3n 3C n3 (1)n C nn 2048.
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: n 11. ....................................................................................................................................
35. Tỡm n tha món C n1 C n2 C nn 1 C nn 4095.
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: n 12. ....................................................................................................................................
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Trang - 140 -
ĐỀ CƯƠNG HỌC TẬP LỚP 11
SỞ GD & ĐT TP. HỒ CHÍ MINH
TRUNG TÂM THẾ VINH – 45 LÊ THÚC HOẠCH – Q. TÂN PHÚ
Mơn: Tốn, Năm học: 2019 – 2020
Dạng tồn chẵn hoặc tồn lẻ
2k 1
Trong biểu thức có C n0 C n2k ... (tồn chẵn) hoặc C n1 C n
... (tồn lẻ) thì đó là
dấu hiệu nhận dạng khai triển hai biểu thức dạng (a b)n và (a b)n khi chọn a, b rồi cộng
lại (khi tồn chẵn) hoặc trừ đi (khi tồn lẻ) theo từng vế.
36. Tìm số ngun dương n thỏa mãn: C 20n C 22n C 24n C 26n C 22nn 512.
Suy luận: Đây là dạng tồn chẵn, sẽ khai triển 2 nhị thức (a b)2n và (a b)2n rồi cộng lại.
Khơng có số mũ của số nào giảm Chọn a 1.
Khơng có số mũ của số nào tăng Chọn b 1.
Lời giải tham khảo
Xét (1 1)2n
2n
C
.12n k .1k
C 20n C 21n C 22n C 23n C 24n C 22nn
(1)
Xét (1 1)2n C 2kn .12n k .(1)k C 20n C 21n C 22n C 23n C 24n C 22nn
(2)
k 0
k
2n
2n
k 0
Lấy (1) (2) 22n 02n 2.(C 20n C 22n C 24n C 26n C 22nn )
22n 2.512 22n 1024 22n 210 2n 10 n 5.
37. Tìm số ngun dương n thỏa mãn: C 21n 1 C 23n 1 C 25n 1 C 22nn11 1024.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
Đáp số: n 5. .......................................................................................................................................
2
4
6
8
1006
38. Tìm số ngun dương n thỏa mãn: C 2014
C 2014
C 2014
C 2014
C 2014
2503n 1.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
Đáp số: n 4. .......................................................................................................................................
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Trang - 141 -
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020
39. Chng minh: C 20n C 22n C 22nn C 21n C 23n C 22nn 1 22n 1.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
0
2
4
100
40. Tớnh tng: S C 100
C 100
C 100
C 100
.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: S 299. ...........................................................................................................................................
0
2
4
2000
41. Tớnh tng: S C 2001
32C 2001
34C 2001
32020C 2001
.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: S 42001 22001. ............................................................................................................................
42.
Tỡm s t nhiờn n tha món: C 20n 32C 22n 34C 24n 32n C 22nn 215 (216 1).
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: n 8. .............................................................................................................................................
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Trang - 142 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
Nhúm bi toỏn tớnh tng hoc chng minh da vo tớnh cht hoc bin i (nõng cao)
43.
Tớnh tng: S
1
1
1
1
2!.2012! 4!.2010!
2012!.2! 2014 !
Suy lun: Da vo cụng thc t hp C nk
n!
, cú: k (n k ) n nờn s phõn tớch
k !.(n k )!
1
1
2
v gi cho ta nhõn thờm hai v cho 2014! s a c v C 2014
.
2!.2012! 2!.(2014 2)!
Li gii tham kho
Ta cú: S
1
1
1
1
2!.2012! 4!.2010!
2012!.2! 2014 !
2014!.S
2014 !
2014 !
2014!
2014!
2!.2012! 4!.2010!
2012!.2! 2014!
2014!.S
2014!
2014!
2014!
2014!
2!.(2014 2)! 4!.(2014 4)!
2012!.(2014 2)! 2014!.(2014 2014)!
2
4
6
2012
2014
2014 !.S C 2014
C 2014
C 2014
C 2014
C 2014
.
Xột ...........................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
22013 1
ỏp s: S
. ..........................................................................................................................
2014 !
44.
Tớnh tng: S
1
1
1
1
2019! 3!.2017 !
2017 !.3! 2020!
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S
22019 1
. ..........................................................................................................................
2020!
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Trang - 143 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
45.
Tớnh tng: S
Ta cú: S
0
C 2013
1
2013
1
1
1
C 2013
2
1
C 2013
2
2
C 2013
3
2013!
2
C 2013
3
2013
C 2013
2014
2013
2013
C 2013
2014
2013
k 0
1
k
C 2013
1k
2013
1
k
C 2013
k 0 1 k
2014.2013!
1 k k !.(2013 k )! 2014 (1 k ).k !.(2013 k )!
k 0
0
C 2013
Mụn: Toỏn, Nm hc: 2019 2020
k 0
2013
1
2014!
2014 (k 1)!.[2014 (k 1)]!
k 0
2013
1
2014 C
k 0
k 1
2014
1
1
2
2014
C 2014
C 2014
C 2014
.
2014
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
22014 1
ỏp s: S
. ..........................................................................................................................
2014
46.
Chng minh: k 2C nk n(n 1)C nk22 nC nk11, vi k, n l s nguyờn tha 2 k n.
1
2
3
2013
Tớnh tng: S 12.C 2013
22.C 2013
32.C 2013
20132.C 2013
.
Ta cú: k 2C nk k .k .C nk k .[(k 1) 1].C nk k (k 1).C nk k .C nk
.............................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
. ................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
ỏp s: S 2013.2014.22011. ...............................................................................................................
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Trang - 144 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
47.
Mụn: Toỏn, Nm hc: 2019 2020
Chng minh: (C n0 )2 (C n1 )2 (C nn )2 C 2nn vi n 2, n .
Suy lun: Ta cú (1 x )2n (1 x )n (x 1)n nờn suy ngh n vic khai trin (1 x )2n v khai
trin tớch (1 x )n .(x 1)n , sau ú so sỏnh h s x n vi nhau s a n pcm.
Li gii tham kho
2n
Xột khai trin: P (x ) (1 x )2n C 2kn x k cú h s ca x n l C 2nn .
k 0
Xột khai trin P (x ) (1 x )n (x 1)n
x n l
n
(Cnk )2 .x n . Suy ra:
k 0
48.
n
(C
k 0
n
n n
k
k
k n k
C
.
x
.
C
x
n n
(C nk )2 .x n cú h s ca
k 0
k 0
k 0 k 0
n
k 2
n
) .x n C 2nn (C n0 )2 (C n1 )2 (C nn )2 C 2nn (pcm).
0
1
2
2020 2
)2 (C 2020
)2 (C 2020
)2 (C 2020
).
Tớnh tng: S (C 2020
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
2020
S: S C 4040 . .......................................................................................................................................
49.
C 0 2 C 1 2
C n 2 C n 1 1
Cho s t nhiờn n 2, chng minh: n n n 2n 2 2
1
(n 1)
2
n 1
2
2
2
2
k
n
n
C 0 C 1
C n
2
C
1
n!
n
n
n
n
Ta cú:
1 2
k 1
k 0
k 0 k 1 k !.(n k )!
n 1
2
1
n !.(n 1)
n
1
k
!.(
k
1).(
n
k
)!
k 0
n
2
1
C nk 11
k 0 n 1
n
Suy ra VT
n
1
(n 1)
k 0
2
1
(n 1)!
n 1 (k 1)!.[(n 1) (k 1)!
k 0
n
2
(C nk 11 )2 .
1
(C 1 )2 (C 1 )2 (C n 1 )2
n 1
n 1
n 1
2
(n 1)
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
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Trang - 145 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
50.
Mụn: Toỏn, Nm hc: 2019 2020
Tỡm s nguyờn dng n tha món: 2C n0 5C n1 8C n2 (3n 2)C nn 1600.
n
Hng dn: 2C n0 5C n1 8C n2 (3n 2)C nn (3k 2)C nk
S: n 7.
k 0
51.
Tớnh tng: S
Hng dn: S
C 1212
11.12
2014
C 1312
11.12
C 1412
11.12
C k12
(k 1).k
k 12
52.
C n102
132
12
C 2013
2012.2013
12
C 2014
2013.2014
S: S
n
n
1
1
C nk
C nk 11.
k 0 k 1
k 0 n 1
11
C 2013
132
1
1
1
1
1023
Tỡm s nguyờn dng n tha món: C n0 C n1 C n2 C n3
C nn
2
3
4
n 1
n 1
Hng dn: VT
53.
S: n 9.
2
3
2013
2.3.C 2013
2012.2013.C 2013
.
Tớnh tng: S 1.2.C 2013
2013
k
k 2
Hng dn: S (k 1).k .C 2013
2012.2013.C 2011
.
S: 2012.2013.22011.
k 2
54.
0
1
2
3
4
2012
2C 2012
3C 2012
4C 2012
5C 2012
2013C 2012
.
Tớnh tng: S C 2012
Hng dn: S
2012
k
(k 1)C 2012
k 0
55.
2010
C
k 0
n
k 0
2010k
k
C 2011
2011C 2010
.
k
Tớnh tng: S
0
A2013
0!
3
k
1
A2013
1!
n
k 3
1
1
1
1
89
3 3 3
3
30
C3 C4 C5
Cn
1
1
3 (k 1)(k 2) k(k 1).
2
A2013
2!
2013
A2013
2013!
Ank
n!
Hng dn: C
v 0! 1.
k !(n k )!
k!
Tỡm s nguyờn dng n 2 tha món:
Hng dn: C nk
59.
S: n 10.
k
n
58.
S: 1007.22012.
S: n 2010.
k
2011
1
C
k 3
k 1
Tỡm s nguyờn dng n 3 tha món:
Hng dn: VT
57.
2012
k 1
k
C 2012
.
2012C 2011
0
2010
1
2009
k
2010k
2010 0
C 2011
C 2011
C 2010
C 2011
C 2011
C 2011
C 1 2011.2n.
Tỡm n tha: C 2011
k
Hng dn: VT
56.
2011
S: S 22013.
1
1
1
1
2013
2 2 2
2
2014
A2
A3
A4
An
Ak
n!
1
1
1 1
1
1
1
n k
VT
k !(n k )!
k!
2 C 22 C 32 C 42
An
k !C nk
C n2
11
1
10 1
11 0
C 20
C1210 C 202 C129 C 20
C 12 C 20
C12 .
Tớnh tng: S C 200 C12
Hng dn: So sỏnh h s x 11 trong (1 x )32 v (1 x )20 (1 x )12 .
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11
.
S: S C 32
Trang - 146 -
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020
BI TP V NH 3
Cõu 1. Xột (1 2x )20 a 0 a1x a 2x 2 a 20x 20 . Giỏ tr a 0 a1 a 2 a 20 bng
A. 1.
B. 320.
..............................................................................................
C. 0.
D. 1.
..............................................................................................
Cõu 2. Tớnh tng cỏc h s trong khai trin (1 2x )2018 .
A. 1.
B. 1.
..............................................................................................
C. 2018.
D. 2018.
..............................................................................................
Cõu 3. Xột khai trin a thc (1 2x 3x 2 )10 a 0 a1x a2x 2 a20x 20 . Giỏ tr ca
tng S a 0 2a1 4a 2 220 a 20 bng
A. 1510.
B. 1710.
..............................................................................................
C. 710.
D. 1720.
..............................................................................................
Cõu 4. Cho a thc P (x ) (x 2)2017 (3 2x )2018 a2018x 2018 a 2017 x 2017 a1x a 0 .
Khi ú S a 2018 a 2017 a1 a 0 bng
A. 0.
B. 1.
..............................................................................................
C. 2018.
D. 2017.
..............................................................................................
1
2
3
2016
Cõu 5. Tng C 2016
C 2016
C 2016
C 2016
bng
A. 42016.
B. 22016 1.
..............................................................................................
C. 42016 1.
D. 22016 1.
..............................................................................................
1
Cõu 6. Tớnh tng S C 100 2.C 10
22.C 102 210.C 1010 .
A. S 210.
B. S 410.
..............................................................................................
C. S 310.
D. S 311.
..............................................................................................
Cõu 7. Tỡm s nguyờn dng n tha món C n0 2C n1 22C n2 2n C nn 14348907.
A. n 15.
B. n 14.
..............................................................................................
C. n 10.
D. n 11.
..............................................................................................
Cõu 8. Tỡm s nguyờn dng n tha: 3n C n0 3n 1C n1 3n 2C n2 (1)n C nn 2048.
A. n 8.
B. n 9.
..............................................................................................
C. n 10.
D. n 11.
..............................................................................................
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Trang - 147 -
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
ẹE CệễNG HOẽC TAP LễP 11
Mụn: Toỏn, Nm hc: 2019 2020
Cõu 9. Tớnh tng S 5n C n0 5n 1.3.C nn 1 32.5n 2C nn 2 3n C n0 .
A. 28n.
B. 1 8n.
..............................................................................................
C. 8n1.
D. 8n.
..............................................................................................
Cõu 10. Tng tt c cỏc h s ca khai trin (x y )20 bng bao nhiờu ?
A. 77520.
B. 1860480.
..............................................................................................
C. 1048576.
D. 81920.
..............................................................................................
Cõu 11. Trong khai trin nh thc (3 0, 02)7 , tỡm tng s ba s hng u tiờn ?
A. 2289, 3283.
B. 2291,1012.
..............................................................................................
C. 2275, 93801.
D. 2291,1141.
..............................................................................................
Cõu 12. Tng C 20n C 22n C 24n C 22nn bng
..............................................................................................
n2
A. 2
n1
.
B. 2
C. 22n2.
.
D. 22n1.
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 13. Tỡm s nguyờn dng n tha món C 21n 1 C 23n 1 C 22nn11 1024.
..............................................................................................
A. n 10.
B. n 5.
C. n 9.
D. n 11.
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 14. Tỡm s nguyờn dng n tha món C 20n 1 C 22n 1 C 24n 1 C 22nn1 1024.
..............................................................................................
A. n 6.
B. n 10.
C. n 5.
D. n 9.
..............................................................................................
..............................................................................................
..............................................................................................
1
3
5
2017
Cõu 15. Tng T C 2017
C 2017
C 2017
C 2017
bng
..............................................................................................
2017
A. 2
C. 22017.
1.
2016
B. 2
.
D. 22016 1.
..............................................................................................
..............................................................................................
..............................................................................................
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Trang - 148 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Cõu 16.
Tng S C n0 .C 21n C n1 .C 22n C nn .C 2nn1 bng
2n
3n
B. C .
C. C 3nn1 .
D. C 32nn 1 .
..............................................................................................
..............................................................................................
..............................................................................................
Tớnh tng P (C n0 )2 (C n1 )2 (C nn )2 theo n.
n
n
..............................................................................................
2
n
A. C .
B. C .
C. C 2nn .
D. C 22nn .
Cõu 18.
..............................................................................................
n
3n
A. C .
Cõu 17.
Mụn: Toỏn, Nm hc: 2019 2020
..............................................................................................
..............................................................................................
..............................................................................................
Tớnh tng S C n1 2C n2 3C n3 nC nn .
A. 4n.2n 1.
B. n.2n 1.
C. 3n.2n 1.
D. 2n.2n 1.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 19.
Tỡm s nguyờn dng n tha món 2C n0 5C n1 8C n2 (3n 2)C nn 1600.
A. n 5.
B. n 7.
C. n 10.
D. n 8.
..............................................................................................
..............................................................................................
..............................................................................................
..............................................................................................
Cõu 20.
1
1
1
Tớnh cỏc tng sau: S1 C n0 C n1 C n2
C nn .
2
3
n 1
2n 1 1
A.
n 1
..............................................................................................
2n 1 1
B.
n 1
..............................................................................................
2n 1 1
2n 1 1
C.
1. D.
1.
n 1
n 1
..............................................................................................
..............................................................................................
P N V NH S 03
1.A
2.B
3.B
4.A
5.D
6.C
7.A
8.D
9.D
10.C
11.B
12.D
13.B
14.C
15.B
16.C
17.C
18.B
19.B
20.B
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Trang - 149 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
Dng toỏn 3. Tỡm h s hoc s hng dng cú iu kin (kt hp gia dng 1 & 2)
n
2
60. Tỡm s hng khụng cha x trong khai trin x 3 2 , bit C n1 C n2 55.
x
iu kin: n 2 v n .
Ta cú: C n1 C n2 55
n!
n!
55 n 2 n 110 0
(n 1)! (n 2)!2!
n 10
n 11 (L).
k
10
2
k 3(10k ) 2
k k 305k
Vi n 10 ta cú x 3 2 vi s hng tng quỏt: Tk 1 C 10
x
.
.
2 C 10 2 x
x
x
S hng khụng cha x ng vi k tha 30 5k 0 k 6.
Vy s hng khụng cha x l C 106 2 6 13440.
n
61. Tỡm s hng cha x
10
1
trong khai trin x 3 2 , bit C n4 13C n2 .
x
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: C104 (1)4 .x 10 210x 10 . ..................................................................................................................
n
62. Tỡm h s ca x
20
1
trong khai trin nh thc Newton x 4 , bit An2 3n 440.
2
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
15 15
2 . .................................................................................................................................
S: (1)15C 20
3
2
1
63. Tỡm s hng cha x 8 trong khai trin (x 2 2)n , bit An 8Cn C n 49.
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 280x 8 . .............................................................................................................................................
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Trang - 150 -
ẹE CệễNG HOẽC TAP LễP 11
S GD & T TP. H CH MINH
TRUNG TM TH VINH 45 Lấ THC HOCH Q. TN PH
Mụn: Toỏn, Nm hc: 2019 2020
n
1
64. Tỡm s hng cha x trong khai trin x 3 2 , x 0, bit Cn0 C n1 C n2 11.
x
2
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 6x 2 . ..................................................................................................................................................
n
2
65. Tỡm h s ca x trong khai trin x 3 , x 0, bit Cnn46 n.An2 454.
x
4
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 1792. ............................................................................................................................................
n
3
66. Tỡm s hng khụng cha x trong khai trin x 3 , bit An21 Cn21 18P3 .
x
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 252. .................................................................................................................................................
n
67. Tỡm h s ca x
11
n
trong khai trin x x n 8 , bit Cnn3 C n21 Cn11C nn32 .
3x
3
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
.................................................................................................................................................................
S: 32440320. .......................................................................................................................................
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Trang - 151 -