数列[an},a1=2,an+1=an+2^n+1,设{bn}满足bn=2log2(an+1-n),证明(1+1/b1)(1+1/b2)……(1+1/bn)对一切n为正

问题描述:

数列[an},a1=2,an+1=an+2^n+1,设{bn}满足bn=2log2(an+1-n),证明(1+1/b1)(1+1/b2)……(1+1/bn)对一切n为正
整数成立

因为a(n+1)=a(n)+2^n+1所以a(n+1)-2^(n+1)=a(n)-2^n+1所以{a(n)-2^n}是等差数列,d=1因为a1-2^1=0所以a(n)-2^n=n-1a(n)=n+2^n+1b(n)=2log2(2^n+2)然后用数学归纳法当n=1,因为左边=1+1/(2log(2)4)=5/4右边=根号2所以原...