A=(1+1/2)(1－1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)……(1+1/m)(1-1/m)B=(1+1/2)(1－1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)……(1+1/n)(1-1/n)若A-B=1/26,且m,n为大于一的正整数,求m,n的值.

Question

A=(1+1/2)(1－1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)……(1+1/m)(1-1/m)
B=(1+1/2)(1－1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)……(1+1/n)(1-1/n)
若A-B=1/26,且m,n为大于一的正整数,求m,n的值.

Answer 1

A=(1+1/2)(1－1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)……(1+1/m)(1-1/m)
=(1+1/2)(1+1/3)(1+1/4)……(1+1/m)(1-1/2)(1-1/3)(1-1/4)……(1-1/m)
=(3/2)(4/3)(5/4)……[(m+1)/m](1/2)(2/3)(3/4)……[(m-1)/m]
=[(m+1)2](1/m)
=(m+1)/2m
B=(1+1/2)(1－1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4)……(1+1/n)(1-1/n)
=(1+1/2)(1+1/3)(1+1/4)……(1+1/n)(1-1/2)(1-1/3)(1-1/4)……(1-1/n)
=(3/2)(4/3)(5/4)……[(n+1)/n](1/2)(2/3)(3/4)……[(n-1)/n]
=[(n+1)2](1/n)
=(n+1)/2n
A-B=1/26
(m+1)/2m-(n+1)/2n=1/26
(m+1)/m-(n+1)/n=1/13
1+1/m-(1+1/n)=1/13
1+1/m-1-1/n=1/13
1/n=1/m-1/13
1/n=(13-m)/13m
n=13m/(13-m)
因为m,n为大于一的正整数,知m=12,n=12×13=156