n n2
n " N n
n+1
T (n) n " N n e"
n0 n0 n0 = 0
T (1), T (2), T (3), T (4), ...
T (n0), T (n0 + 1), T (n0 + 2), T (n0 + 3), ...
n " N
T (1), T (n), ...
T (1)
n T (n)
T (n + 1)
T (n) =Ò! T (n + 1),
n " N T (n)
T (1), T (2), ...
T (n)
T (2) T (2)
 T (3)
T (3) T (4)
n T (n)
 n n2
n2
12
n2
(n + 1)2
L = 1 + 3 + 5 + ... + (2n - 1) + (2(n + 1) - 1)
= 1 + 3 + 5 + ... + (2n - 1) + (2n + 1)
= n2 + 2n + 1
= (n + 1)2
= P
"n e" 1 n
n2
n " N
n

n(n + 1)
i =
2
i=1
n(n+1)
2
T (n)
T (1)
1

1(1 + 1)
i = = 1,
2
i=1
n

n(n + 1)
i =
2
i=1
n+1

(n + 1)(n + 2)
i = .
2
i=1
n+1 n

n(n + 1)
L = i = i + (n + 1) = + n + 1
2
i=1 i=1
n(n + 1) + 2(n + 1) (n + 1)(n + 2)
= =
2 2
= P

 n k = 0, ..., n

n n!
= .
k k!(n - k)!

n n!
= .
k k!(n - k)!
1 1
1!
= 1 = 1. = 1
0 1 0!(1-0)!
1!
= 1,
1!(1-1)!
n
n!
=
k k!(n-k)!
n+1
(n+1)!
=
k k!(n-k+1)!
n+1
k

n + 1 n n n! n!
= + = +
k k - 1 k (k - 1)!(n - k + 1)! k!(n - k)!
n!(n - k + 1) + n!k
=
k!(n - k + 1)!
k! = (k - 1)!k
(n - k + 1)! = (n - k + 1)!(n - k)
n!(n - k + 1 + k) n!(n + 1)
= =
k!(n - k + 1)! k!(n - k + 1)!
(n + 1)!
=
k!(n - k + 1)!

 f(x) = x
(xn) = nxn-1 n " N
(x) = 1 = 1 · x1-1,
(xn) = nxn-1
(xn+1) = (n + 1)xn-1+1 = (n + 1)xn
(xn+1) = (xnx) = (xn) x + xn(x)
= nxn-1x + xn = nxn + xn
= (n + 1)xn.

n " N
3
10n + 4n - 2
 10n + 4n - 2 = 3a a " N
3
10n+1 + 4n+1 - 2
10n+1 + 4n+1 - 2 = 10n · 10 + 4n · 4 - 2
= 10n · (9 + 1) + 4n · (3 + 1) - 2
= 10n · 9 + 10n + 4n · 3 + 4n - 2
= 10n + 4n - 2 + 10n · 9 + 3 · 4n
= 3a + 10n · 9 + 3 · 4n
= 3(a + 10n · 3 + 4n)

q = 1

1 - qn+1
a1 + a1q + a1q2 + ... + a1qn = a1 .
1 - q
a1 = a1
a1 + a1q + a1q2 + ... + a1qn = a1 1-qn+1
1-q
a1 + a1q + a1q2 + ... + a1qn + a1qn+1 = a1 1-qn+2 .
1-q
L = a1 + a1q + a1q2 + ... + a1qn + a1qn+1
1 - qn+1
= a1 + a1qn+1
1 - q

1 - qn+1
= a1 + qn+1
1 - q

1 - qn+1 qn+1 - qn+2
= a1 +
1 - q 1 - q
1 - qn+2
= a1
1 - q
= P
q = 1

T (1), T (2), ...
T (1), T (2), ...
T (1), T (2), ...,
T (1), T (n), ...
 T (1)
n " N n
T (1), T (2), ..., T (n)
T (n + 1)
T (1), T (2), ..., T (n) =Ò! T (n + 1),
n " N T (n)
T (1), T (2), T (3), ...
T (1)
T (2),
T (1), T (2)
T (3)
T (1), T (2), T (3) T (4),
n T (n)
n
T (1), ..., T (n)
A ‚" N, A = " =Ò! "n0 " A "n " A n e" n0

k0
k0
A.
k = k0 - 1
k0,
T (k0) = T (k+1)
k0.
 k0
k0
k = k0 - 1
k0
T (k0) = T (k+1)
k0.
4n - 1
3, 15, 63, 255, ...
4n - 1
 n0 41 - 1 = 3.
4n - 1
4n - 1 = 3a, a " N
4n+1 - 1
4n+1 - 1 = 4 · 4n - 1 = (3 + 1)4n - 1
= 3 · 4n + (4n - 1) = 3 · 4n + 3a
= 3(4n + a).

2n > 10n
2 < 10, 4 < 20, 8 < 30, 16 < 40, 32 < 50, 64 > 60, 128 > 70.
n e" 6
2n > 10n 26 = 64 > 10 · 6.
2n > 10n n e" 6
 2n+1 > 10(n + 1) n e" 6
2n+1 = 2 · 2n = 2n + 2n > 10n + 10n > 10n + 10
= 10(n + 1).
n e" 6
an
Å„Å‚
òÅ‚
a1 = 1,
.
ół
an+1 = 2an + 1 n e" 1
an.
a1 = 1, a2 =
2a1 + 13, a3 = 2a2 + 1 = 7, a4 = 2a3 + 1 = 15.
2, 4, 8, 16, ..., 2n,
an = 2n - 1.
an
n e" 1.
a1 = 21-1 = 1.
 an = 2n - 1
an+1 = 2n+1 - 1
an an+1 = 2an + 1,
an = 2n - 1,
an+1 = 2an + 1 = 2(2n - 1) + 1 =
= 2 · 2n - 2 + 1 = 2n+1 - 1.

an
a1 = A, a2 = 2A an+1 = 2an - an-1 A, n " Z
a1 = A a2 = 2A
an-1 = A(n - 1), an = nA
an+1 = A(n + 1)
an
an+1 = 2an - an-1 = 2(nA) - A(n - 1) =
= 2nA - (nA - A) = 2nA - nA + A
= A(n + 1)
> 0
an
n
n
n + 1
n
n + 1
n
n + 1
 n