− − − − −⋆ −−−−−
class="bi x0 y2 w2 h3"
class="bi x0 y2 w2 h4"
class="bi x0 y2 w3 h5"
class="bi x0 y2 w3 h4"
u
n
u
1
, u
2
, u
3
,
u
1
, u
2
, u
3
,
u
i
i
u
1
, u
2
, u
3
,

m u
n
> m n = 1, 2,
u
1
, u
2
, u
3
, N
o
u
n
= C n ≥ N
o
C
u
1
, u
2
, u
3
, n
k p = 1, 2,







} : u
1
, u
2
, u
3
, ··· {v
n
} : v
1
, v
2
, v
3
, ···
{u
n
+ v
n
} : u
1
+ v
1
, u
2
+ v
2
, u
3
+ v

1
v
1
,
u
2
v
2
,
u
3
v
3
···).
{u
n
} u
n
= 2n + 1
n = 0, 1, 2, ··· 1, 3, 5, 7, ···
u
0
u
0
, u
1
, u
2
, ···
{u

j
v
j
k + v
j−1
n u
j
v
j
k = nu
1
+ v
1
;
k + v
1
= nu
2
+ v
2
;
k + v
2
= nu
3
+ v
3
;
···
k + v

1 f i = 1, 2, ···
∆
2
y
i
= ∆y
i
−∆y
i−1
= (y
i
−y
i−1
) −(y
i−1
−y
i−2
) = y
i
−2y
i−1
+ y
i−2
2 f i = 1, 2, ···
•
∆
k
(f ± g) = ∆
k
(f) ± ∆

.
•
y = f(x)
x
0
, x
0
+ h, x
0
+ 2h, , x
0
+ nh,
y
o
, y
1
, y
2
, , y
n
,
a
n
y
n+i
+a
n−1
y
n−1+i
+a

,
y
i
, y
i+1
, , y
i+n
y
′
i
, y
′
i+1
, , y
′
i+n
y
i
± y
′
i
, y
i+1
± y
′
i+1
, ··· , y
i+n
± y
′

, , λ
n
n
a
n
λ
n
+ a
n−1
λ
n−1
+ ··· + a
1
λ + a
0
= 0.
λ
1
s
y
i
= c
1
λ
i
1
+ c
2
iλ
i

= −1; u
2
= −1; u
3
= 1; u
4
= 5;
u
5
= 11; u
6
= 19; u
7
= 29; u
8
= 41; u
9
= 55.
n = 0, 9.
y = f(x)
∆y
∆
2
y
ax
2
+ bx + c x
a, b, c x = 0, 1, 2
u
0

x
2
− 3x + 1
u
n
= n
2
− 3n + 1, n = 0, 9.
{u
n
}




















= 2
u
n
u
n
= c
1
λ
n
1
+ c
2
λ
n
2
u
n
= c
1
+ c
2
2
n
.
c
1
c
2
u
0

u
n
= 2
n
u
1
, u
2
, u
3
, d (d = 0)
u
n
= u
n−1
+ d n = 2, 3,
•
u
n
= u
1
+ (n − 1)d, n = 1, 2, 3,
u
k
=
u
k−1
+ u
k+1
2

n−k
k = 2, 3, , n −1.
•
u
1
, u
2
, d
S
n
= u
1
+ u
2
+ ··· + u
n−1
+ u
n
.
S
n
=
(u
1
+ u
n
)n
2
=
[2u

1
=
n(n + 1)
2
S
2
=
n(n + 1)(2n + 1)
2
;
S
3
=
n
2
(n + 1)
2
4
.
u
1
, u
2
, u
3
, q (q = 0, q = 1)
u
n
= u
n−1

+ ··· + u
n
.
S
n
=
u
1
(q
n
− 1)
q −1
.
u
1
, u
2
,













√
5
2
λ
2
=
1 −
√
5
2
.
u
n
= c
1
λ
n
1
+ c
2
λ
n
2
= c
1

1 +
√
5
2

2
+ c
2
1 −
√
5
2
u
2
= 1 = c
1

1 +
√
5
2

2
+ c
2

1 −
√
5
2

2




√
5
2

2
= 1
⇒







c
1
=
1
√
5
c
2
= −
1
√
5
u
n
=
1

3
+ u
5
+ ···+ u
2n−1
= u
2n
u
2
+ u
4
+ u
6
+ ···+ u
2n
= u
2n+1
− 1
u
2
1
+ u
2
2
+ ··· + u
2
n
= u
n
u

n−1
u
n+1
= (−1)
n+1
a b a b a b
b a k a = k.b a b
a  b a b
p > 1
p.
a, b a b a ≥ b
a
i
b i = 1, n (a
1
+ a
2
+ ···+ a
n
) b
a b b = 0
q r a = bq + r 0  r < b.
a b
a b (a, b)
(a, b) a b
a b (a, b)
[a, b] a b
n a
1
, a

1
, a
2
, , a
n
• b a
i
, ∀i = 1 , n
• b
′
b a
i
, ∀i = 1, n b
′
b.
b = [a
1
, a
2
, , a
n
].
a b (a, b) = (a, a + b)
m
(ma, mb) = m(a, b);
[ma, mb] = m[a, b].
(a, b) d

a
d

(i = 1, k) p
i
(i = 1, k)
1 < p
1
< p
2
< ··· < p
k
n
a, b p ab p
a p b p
a b m (m = 0)
a b m a ≡ b ( mod m).
a b m m
(a − b) m.
Z.
a ≡ b (mod m) c ≡ d (mod m)
a + c ≡ b + d (mod m),
a − c ≡ b −d (mod m),
ac ≡ bd (mod m).
p ab ≡ 0 (mod p) a ≡ 0 (mod p)
b ≡ 0 (mod p)
p a
(a
p
− a) ≡ p (a, p) = 1 a
p−1
≡ 1 (mod p)
m (a, m) = 1 a

n n[x]  [nx]
n q q = 0 q

n
q

 n.
{u
n
} n = 0, 1, 2,
u
1
= 1;
u
m+n
+ u
m−n
=
1
2
(u
2m
+ u
2n
), ∀m ≥ n; m, n ∈ N
u
0
+ u
0
=

2
(
u
0
= 0).
u
1
= 1 u
2
= 4 u
0
= 0
2
, u
1
= 1
2
, u
2
= 2
2
u
n
= n
2
n = k
m = k, n = 0
u
k
+ u

).
u
k+1
=
1
2
u
2k
+
1
2
u
2
− u
k−1
u
2k
= 4 k
2
u
2
= 2
2
, u
k−1
= (k − 1)
2
u
k+1
=

u
1
= 1; u
2
= −1
u
n
= −u
n−1
− 2u
n−2
; n ≥ 3.
a = 2
2006
− 7u
2
2004
v
n
= 2
n+1
− 7u
n−1
a = v
2005
= 2
2006
− 7u
2
2004

)
2
n = 2 n = k (k ≥ 2),
v
k
= (2u
k
+ u
k−1
)
2
.
n = k + 1
v
k+1
= 2
k+2
− 7u
2
k
.
{u
n
}
(2u
k+1
+ u
k
)
2

u
k−1
+u
2
k−1
)+14u
2
k−1
−7u
2
k
= 2(2u
k
+u
k−1
)
2
+14u
2
k−1
−7u
2
k
=
= 2v
k
+ 14u
2
k−1
−7u

n = k + 1
∀n = 2, 3,
{v
n
}
a = 2
2006
− 7u
2
2004
{u
n
}





u
1
= 1; u
2
= 2
u
n+1
= u
n
(u
n
− 1) + 2; n = 2, 3,

+ 1) −1 = u
2
1
= 1
2
= (2 −1)
2
= (u
2
− 1)
2
.
k = 1
k = n s
n
= (u
n+1
− 1)
2
k = n + 1
s
n+1
= (u
2
1
+ 1)(u
2
2
+ 1) (u
2

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

k = n + 1
k = 1, 2, u
1
, u
2
u
n
n. s
k
k = 1, 2, {s
n
}
{u
n
}





u
0
= 3, u
1
= 17
u
n
= 6u
n−1
− u

+ b(3 −
√
8)
n
.
u
0
= 3, u
1
= 17 a, b





a(3 +
√
8)
0
+ b(3 −
√
8)
0
= 3
(3 +
√
8)
1
+ b(3 −
√

(3 −
√
8).
u
n
=
1
2

(3 +
√
8)
n+1
+ (3 −
√
8)
n+1

.
v
n
=
1
2
(u
2
n
− 1) =
1
2

2

2
.
(3 ±
√
8)
n+1
= M ±N
√
2, M, N
1
2
(u
2
n
− 1) = N
2
{a
n
}





a
0
= 1, a
1

+ β(2 −
√
3)
n
(α, β ∈ R).
a
0
= 1, a
1
= 2 α, β





α + β = 1
2(α + β) +
√
3(α − β) = 2
⇔ α = β =
1
2
.
a
n
=
1
2
[(2 +
√

(
√
3 − 1)
2
=

√
3 −1
√
2

2
.
a
n
− 1 =
1
2


√
3 + 1
√
2

2n
+

√
3 −1

3 + 1)
n
− (
√
3 − 1)
n
(
√
2)
n+1
∈ Z.
n = 0 ⇒ A
1
= 0 ∈ Z.
n = 1 ⇒ A
2
= 1 ∈ Z.
n = 2k, k ∈ N
∗
,
(
√
3 + 1)
n
− (
√
3 − 1)
n
(
√

k
b
1
=
√
6 ⇒ b
k
∈ Q, ∀k ∈ N
∗
a
n
− 1
n = 2k, k ∈ N
∗
n = 2k + 1, k ∈ N
∗
(
√
3 + 1)
n
− (
√
3 −1)
n
(
√
2)
n+1
=
√

].
c
k
=
√
3 + 1
2
[(2+
√
3)
k
−(2−
√
3)
k
], k ∈ N
∗
{c
k
} c
k+2
= 4c
k+1
−c
k
c
1
= 5 ⇒ c
k
∈ Z, ∀ k ∈ N

n+2
= 2u
n+1
−u
n
+1 u
n+2
−u
n+1
= u
n+1
−u
n
+1.
u
n+2
− u
n+1
= v
n+2
v
n+2
= v
n+1
+ 1
d = 1.
u
n
= (u
n

+ 1 =
n(n + 1)
2
.
u
n
=
n(n + 1)
2
n = 1, 2,
A
n
= 4
n(n + 1)
2
·
(n + 2)(n + 3)
2
+ 1 = n(n + 1)(n + 2)(n + 3) + 1 = (n
2
+ 3n + 1)
2
.
A
n
n
{u
n
}
u

− u
n
− u
n−1
+ 2u
n
.
v
n
= u
n+2
− u
n+1
− u
n
, n = 1, 2, v
n
= v
n−1
+ 2u
n
, ∀n = 2, 3,
∀n = 2, 3,