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数学归纳法及其应用举例(2)

目 的 要 求

1.会用数学归纳法证明整数(整式)整除问 题. 2.会用数学归纳法证明一些简单的几何 问题. 3.了解数学归纳法应用的广泛性,进一 步掌握数学归纳法的证明步骤. 4.掌握为证n=k+1成立的常见变形技巧: 提公因式、添项、拆项、合并项、配方等

复 习 旧 课 , 提 出 任 务

①数学归纳法证明有哪些步骤?

②数学归纳法通常解决什么问题?
(与正整数有关命题)

例 题 选 讲

例1 用数学归纳法证明: 34n+2+52n+1能被14整除.

证明:(i)当n=1时,34×1+2+52×1+1=754=14×16, ∴当n=1时,34n+2+52n+1能被14整除. (ii)设n=k(k≥1,k∈N*)时,34k+2+52k+1能被14整 除. 那么当n=k+1时

34(k+1)+2+52(k+1)+1=34k+2· 34+52k+1· 52 =81· 34k+2+25· 52k+1 =(25+56)· 34k+2+25· 52k+1

=25· (34k+2+52k+1)+56· 34k+2.

∵ (34k+2+52k+1)能被14整除,56能被14整除,

例 题 选 讲

∴ 34n+2+52n+1能被14整除.即n=k+1时, 命题成立. 根据(i)、(ii)可知, 34n+2+52n+1能被14整除. 例2:用数学归纳法证明: x2n-y2n能被x+y整除. 例3 平面内有n(n≥2)条直线,其中任何

两条不平行,任何三条不过同一点,证明交点 1 的个数为: f (n) ? n(n ? 1) 2

n

图形

f(n)
f(1)=0 f(2)=1=f(1)+1 f(3)=3=f(2)+2 f(4)=6=f(3)+3

例 题 选 讲

1 2 3

4
… k K+1 …

… f(k) f(k+1)=f(k)+k

例4 已知f(x) ?

xn ? x? n x n ? x ?n

, g(n) ?
2 x ?1
2n

n 2 ?1 n 2 ?1

(n ? N )
*

例 题 选 讲

试比较f( 2 )与g(n)的大小, 并说明理由.

分析 f ( x) ?

x 2 n ?1 x 2 n ?1

? 1?

, f(

2 2) ? 1? n , 2 ?1

比较f ( 2 )与g (n)的大小既比较2 n 与n 2的大小。

例5 设

a n ? 1 ? ? ? ?? (n ? N ),
1 2 1 3 1 n

?

是否存在

n 的整式 g (n) , 使得等式
n 都成立?

a1 ? a 2 ? a3 ? ? ? a n ?1 ? g (n) ? (a n ? 1)

对大于1的一切自然数 并证明你的结论.

当 n =2 时,由 a1 ? g (2)[a2 ? 1] ? g (2) ? 2

例 题 选 讲

当 n =3 时,由 a1 ? a2 ? g (3)[a3 ?1] ? g (3) ? 3 猜想:g( n )= n ( n ? 2), 用数学归纳法证明(略)

研 究 题

研究题: 1.平面内有n条直线,任意两条不平行, 任意三条不共点,求它们彼此分成的线段 数H(n).(H(n)=n2,证明略) 2.对n∈N*,n3+5n+6能被6整除吗?为 什么? (能.用数学归纳法证明略)


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