Bồi dưỡng toán 11 nâng cao phần dãy số

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nn
+
=
+1
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()
(
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dnUU
n
1
1
−
+
=
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2
11 +−
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nn
n

UU
U  
(
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(
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[
]
2
12
2

1
1
21
ndnU
nUU
UUUS
n
nn
−+
=
+
=+++=
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qUU
nn
=

+1
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()
1
1
−
=
n
n
qUU  
11
2
+−
=
nnn
UUU

()
(
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1
1

1
21
−
−
=+++=
q

qU
UUUS
n
nn
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81
=
=
UU 
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nnS
n
65
2
+=
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n
S ,12
=>?97@/
n
U 812AB$C
1
−
−
=
nnn
SSU #$DE$.F G

(
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(
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[
]
101110110
1
=
+
−
−
+
=
−
−
nnUU
nn
#1
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=>X WY12WZS!,
(
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n
a
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1 2 3 4
1 2 3
40
104
n n n n
a a a a
a a a a
− − −
+ + + =
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
+ + + =


] ^_/

1 2 1 3 2 4 3
n n n n
a a a a a a a a
− − −
+ = + = + = +
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(
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1
4 144
n
a a+ =
1
36
n
a a
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1
216 12
2
n
n
a a
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+
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1

1
4 6 40
4 38 104
a d
a d
+ =
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1
7
2
a
d
=
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⇔
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=
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H-
(
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n
a
÷
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(

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1
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a
=
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2
d
=

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+, INS!/ /48
α
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2
1 sin ;sin ;1 sin 3
α α α
+ +
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?6
α
#Da
1
sin
2
α
−

= 
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2 5
n
a n
= −
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&D

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1
2
n n
a a
−
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20
320
s
=
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$UIDbNS!/0<
X" RWZS!,
; 2; 4; 6
a a a a
+ + +
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(

)
(
)
(
)
2 4 6 19305
a a a a+ + + =
(
)
(
)
2 2 2
6 6 8 19305
a a a a + + + =
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2
6
u a a
= +
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135; 134
u u
= = −
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•
H 
2
6 135 9 15

a a a a
+ =
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H 
2
6 134
a a+ = −
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1
n
a
m
=
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1
m
a
n
=
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(
)
m n
≠
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812

=>N<
1
1
a d
mn
= =
1
1
2 2
mn
mn
a a
mn
S mn
+
+
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1 2
; ;
n
u u u
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0 1,
i
u i n
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1 2 2 3 1 1
1 1 1 1

. .
n n n
n
u u u u u u u u
−
−
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1 2 2 3 1 1
1 1 1 1

n n n
n
u u u u u u u u
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−
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1 2 1 1 2 1 1 1 2
1 1 1 1 2 1 1 1

n n n n n n
u u u u u u u u u u u u u
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+
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1 2 2 3 1
1 1 1

. .
n n
u u u u u u
−
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1 2 2 3 1

. .
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n n
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u u
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−
−
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1
1 2 2 3 1 1 1
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n
n n n n
u u
u u u u u u u u u u
−
−
= − + − + + − = − =
(
)
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1
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n n
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n
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u u u u
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n
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2 1
2 1 3 2 1

n n
n n
u u u u
u u
S
u u u u u u

−
−
− −
−
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n n
u u u u u u d
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(
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1
1
1
1 1
1
1
n
n
n
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
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( )
1
1 2 1 1 2 1 1 2
1 1 1 1 1 1 1
( ) 2
n
n n n n n
u u
u u u u u u u u u u u
− −
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+ + + + + = + + +
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X" 
S
=
( )
1
1 2 1 1 2 1
1 1 1 1
( )
n
n n n n
u u
u u u u u u u u
− −
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1 2 1 1 2 1

n n n n
u u u u u u u u
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+ = + = = + = +
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1 2 1 1 2 1
1 2 1 1 2 1
( )
n n n n
n n n n
u u u u u u u u
u u u u u u u u

− −
− −
+ + + +
+ + + + 
1 2 1 1 2 1
1 1 1 1 1 1 1 1

n n n n
S
u u u u u u u u
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#
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1 1 1
2
n
u u u
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+ + +
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 
H-A(C0c
+, -! 4546!" <
(
)
3 2
3
n n n
S S S

= − 
N--He#
(
)
2
3
n n
S S
−
(
)
(
)
(
)
(
)
1 1
2 2 1 2 2 1
3
2 2
u n d n u n d n
 
+ − + −
= −
 
 
 
#
(

)
1
3
2 3 1 3
2
n
u n d n
S
+ −
 
 
= 
+, -662!2O
m n
S S
=
 
m n
≠
! 45
0
m n
S
+
=

=>Nf W 
(
)
(

)
1 1
2 1 2 1
2 2
m n
u m d m u n d n
S s
+ − + −
   
   
=  =
(
)
(
)
2 2
1 1
2 2
u m m m d u n n n d

+ − = + − A(C
(
)
(
)
1
2 1 0
m n u m n d

− + + − =

 
 
>6
m n
≠
;fA(C4
(
)
1
2 1 0
u m n d
+ + − =
A((C
] ^_/
(
)
(
)
1
2 1
2
m n
u m n d m n
S
+
+ + − +
 
 
= H-
0

m n
S
+
=

+, -N7890: ;812,
3 1
n
n
S
= −
! 4512,12
*
=>NfO7812
(
)
(
)
1
1
3 1 3 1
n n
n n n
a S S
−
−
= − = − − −
1 1
3 3 2.3
n n n

n
a
− −
 = − =

.FJK
1
2
1
3.3
3.3
n
n
n
n
a
a
−
−
−
= #I ,g
{
}
n
a
,*<3 @#I
+, -S!I<735$&R,I90: ; 8!*,0hi ,/
9I%$I%$b8!
=>X WY
1 3 2 13 3 15

; ;
u a u a u a
= = =

[ 0<
1 1 2 1 3 1
2 ; 12 ; 14
u a d u a d u a d
= + = + = +
H 1, 8
N\6 W 
1 2 3
124
u u u
+ + =
(
)
1
3 28 124 1
a d + = 
] ^_/N\6O8*
( ) ( )( ) ( )
2
1 1 1 1
12 2 14 2 29 0 2
a d a d a d a d+ = + + ⇔ + =

NfA$C,A&C4
1
116; 8

a d
= = −
H-
1 2 3
100; 20; 4
u u u
= = =

+, -63-,*Lc4fJK6&%$aQa&Qa-0J
j!=2S!/0<
=>X" /WZS!,
2 3
; ; ;
a aq aq aq

[ 0<\6 W &a@&a@
&
kQa@
I
k&Q-,>60<
(
)
(
)
(
)
2
2 1 2 7
aq a aq
− = − + −






(
)
(
)
(
)
2 3
2 7 1 27
aq aq aq− = − + −
?4
( )
( )
2
2
1 7
1 14
a q
aq q

− =


− =



7
2
a
q
=

⇔

=


H-3WZS!,
+, -N7/98!*V 935bT<73SJK8/
98<35RRl=2S!,
=>N\6 W 
1
1 1 1
56
1
n
a
a a q a q
q
+ + + = =
−
,
2
2 2
2 2
1

1 1 1
2
448
1
n
a
a a q a q
q
+ + + + = =
−

[ 0<<
( )
1
2
2
1
56(1 )
448 1
a q
a q
= −



= −


(
)

1
8 1
a q
 = +
>60<
( ) ( )
3
8 1 56 1
4
q q q
+ = −  =
,
$
#$R
+, -+<735Tb-,!*mL3!0KM,
3$U0KM30Jn!NS!I0<
3NS!3-,*3 4578c35&TP7/3SJK8
c35ITR
?ba$baRb
+$laTa&6^&aTa$l
+, -6*
1 2
; ;
u u ! 45
2
2 3 2
n n n
n n n n
S S S
S S S S

−
=
− −

N--HN#
(
)
( ) ( )
1
2
2
1 1
1
1
1
1 1
1 1
1 1
n
n
n n
n n
u q
q
q
q q
u q u q
q q
−
−

−
=
− − +
− −
−
− −
#
( )
1 1
1
n
n
n n
q
q
q q
−
=
−
A(C
He#
(
)
(
)
( ) ( )
2
1 1
2
3 2

3 2
1 1
1 1
1
1 1
1 1
n n
n n
n n
n n
u q u q
q q
q
q q
u q u q
q q
− − −
−
−
=
−
− −
−
− −
#
( )
2
2
1
n n

n
n n n
q q
q
q q q
−
=
−
A((C
H-fA(C,A((C4e]
+, -! 45 ; ;
2 2 2
A B C
tg tg tg
N\6o-,S6pa6+a6
q\6o-,
=>N\6 W 
sin sin
2 2
2 2
2 2 2
cos cos cos
2 2 2
A C B
A C B
tg tg tg
A C C
+
+ = ⇔ =
2

cos 2sin cos cos
2 2 2 2
B B A C
⇔ = 
1 cos 1 cos cos cos
B B A C
⇔ + = − + +
2cos cos cos
B A C
⇔ = +

+, -! 45 ; ;
2 2 2
A B C
cotg cotg cotg
N\6o-,S39a
3aq\6o-,




=>N\6 W 
sin cos
2 2
cot cot 2cot 2
2 2 2
sin sin sin
2 2 2
A C B
A C B

g g g
A C B
+
+ = ⇔ =
2sin
2
cos
2
A C
A C
+
=
+
sin cos sin cos cos
2 2 2 2 2
A C A C A C A C A C
+ + + − +
 
⇔ = −
 
 
( ) ( ) ( )
1 1
sin sin sin sin sin sin
2 2
A C A C B A C
⇔ + = + ⇔ = +
2 ; ;
a c b a b c
⇔ + = ⇔ ÷


+, -! 45 ; ;
2 2 2
A B C
cotg cotg cotg
N\6o-,S39
2 2 2
; ;
a b c
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(
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sin
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sin .sin sin
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2 ; ;
b a c a b c
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