º º
December 26, 2004
1
1
1.1
. . . . 1
1.2
. . . . 2
1.3
. . . . 3
1
ºDZ
n = 1
n = k
n = k + 1
n = 1
n = 1, 2, 3, · · · , k
n = k + 1
-
a 1 + a 2 + · · · + a n
n ≥ √ n
a 1 a 2 · · · a n
1.1
DZ
n
n
n
n
! "
5
2 2 × 3 = 6
1 1 2 1 1 × 1 = 1 2 × 1 = 2
3
1 1 1 × 1 = 1
#
6 + 1 + 2 + 1 = 10
$
n = 1
Æ0
n = 2
1
n = 3
3
DZ
n
n(n − 1) 2
(1) n = 1, 2, 3
(2)
n = 1, 2, 3, · · · , k − 1
n(n−1) 2
n = k
a
k − a (1 ≤ a ≤ k − 1)
a · (k − a)
a
a(a−1)
2
k − a
(k−a)(k−a−1)
2
a(k − a) + a(a − 1)
2 + (k − a)(k − a − 1)
2 = k(k − 1)
2 .
n(n−1)
2
%
21
$1.2
1/3, 1/11, 1/231
1
"
2 3 = 1
2 + 1 6 , 3
7 = 1 3 + 1
11 + 1 231 , 8
11 = 1 2 + 1
6 + 1 22 + 1
66
a/b
a, b
1 < a < b
&' (
1880
Æ
1.1
a/b
(
a
(1) a = 1
a/b = 1/b
(2)
< a(a ≥ 2)
a/b
(
1 > a/b > 0
1 > 1/2 > 1/3 > · · · > 0
q(q ≥ 2)
1 q < a
b < 1 q − 1 .
0 < aq − b < a, a b = 1
q + aq − b
bq
aq − b bq < 1
q .
aq − b bq
( &
1/q
a/b
(
1.3
1.2 (
)
2n (n ≥ 2)
&
n 2 + 1
n 2 + 1
'
(1) n = 2
4 = 2 · 2
5 = 2 2 + 1
)* +
(2)
2(n − 1)
(n − 1) 2 + 1
,!
2n
n 2 + 1
n 2 + 1 ≥ 1
"P, Q
2n
&
P Q
(i)
2(n − 1)
R
P, Q
P QR.
(ii)
2(n − 1)
P, Q
2(n − 1)
P, Q
2(n − 1)
2(n − 1)
≥ n 2 + 1 − 1 − 2(n − 1) = (n − 1) 2 + 1
2(n − 1)
1.1
#1 3 + 2 3 + 3 3 = 9 × 4, 2 3 + 3 3 + 4 3 = 9 × 11, 3 3 + 4 3 + 5 3 = 9 × 24.
1.2
#a n
a 1 = 3, a n+1 = a n (a n + 2) (n ≥ 1)
(1)
a n
(2)
1.3
n
$1 + 3 + 5 + · · · + (2n − 1)
(2n + 1) + (2n + 3) + (2n + 5) + · · · + (4n − 1)
-
1.4
< a n >
#&n i=1
a 3 i =
n
i=1
a i
2
, n ≥ 1.
#
a n = n
1.5
#< f n >
(a) f n
&f 2 = 2
(b)
a < b
f a < f b
(c)
a
b
f a·b = f a · f b
n
f n = n
1.6
,
%& '( ,
)
%&
1.7
(1)
97 4
((2)
n
n 2
(3)
n
8|(n − 5)
4 n
1.8
n
*n 2 + n + 17
" .
/
n
DZ
1 · 2 · 3 · 4 + 1 = 5 2 , 2 · 3 · 4 · 5 + 1 = 11 2 , 3 · 4 · 5 · 6 + 1 = 19 2 .
) DZ
1
,)1n(n + 1)(n + 2)(n + 3) + 1 = 2 .
n
2
n! + 1
n = 4 ⇒ 4! + 1 = 5 2 , n = 5 ⇒ 5! + 1 = 11 2 , n = 7 ⇒ 7! + 1 = 71 2 .
"3
n! + 1
)1 +
4 -DZ
1889
! +,
"#
- 1
5 #
n ≥ m ≥ 1
m j=0
m j
2
n + 2m − j 2m
=
n + m m
2 .
- 34.6 /
P.
Turan
7$ DZ #08
/1
9 " :+
