1
Metody dowodzenia
twierdzeń
p, q, r, . . .
'" (" Ò!
Ô! Ź
p Ò! q
p q p q q
p p
q p Ô! q p
q
p Ò! q p
q
p q
a a - 4
a3 + 1
x x2 - 3x - 10 = 0
x = -2 x = 5
(p Ò! q) Ô! (Źq Ò! Źp).
q p
a b
a b
" " n
a b a n b n
(p Ò! q) Ô! (Źp (" q).
(p Ò! q) Ô! Ź(p '" Źq) ,
p q
(p '" Źq)
p(n) n
p(n)
n n n0
p(n0)
k n0
p(k) Ò! p(k + 1),
p(k + 1) p(k)
p(k)
p(k + 1)
n
n
13 + 23 + . . . + n3.
n
6n+2 + 72n+1
43
 n 4
3n > n3.
n
a d
1
n (2a + (n - 1)d) .
2
p(n) n
p(n)
n n n0
p(n0)
k n0
(p(n0) '" p(n0 + 1) '" · · · '" p(k)) Ò! p(k + 1)
p(k+1) p(i)
n0 i k
a0 = 12, a1 = 29
n 2
an = 5an-1 - 6an-2
n
an = 5 · 3n + 7 · 2n .
n m n > m
n m n > k · m k " N
k + 1
n
 4
n + 1
n
a b a + b
n n2
n
n n > 1 n
" n
p p d" n
n
n(n+1)(2n+1)
12 + 22 + 32 + . . . + n2 =
6
13 + 33 + 53 + . . . + (2n - 1)3 = n2(2n2 - 1)
n(n+1)(n+2)(n+3)
1 · 2 · 3 + 2 · 3 · 4 + . . . + n · (n + 1) · (n + 2) =
4
n " N
1 · 1! + 2 · 2! + 3 · 3! + · · · + n · n! = (n + 1)! - 1.
n 0
11n+2 + 122n+1
133
n
a q q = 1

a(1 - qn)
.
1 - q
a0 = 6, a1 = 11 n 2
an = 3an-1 - 2an-2 ,
n 0
an = 5 · 2n + 1 .
n × n -1, 0, 1
 n+1
2n 2n + 1
n+1
2n
n
n
A {1, 4, 7, 10, 13, . . .
100} A 104
n A n + 1
[2n] A a b a b