Question

用数学归纳法证明：1+1/3+1/5+1/(1+2+3+n)=2n/(n+1)

Answer 1

1、n=1时,左边=1,右边=2×1/（1+1）=1,故命题成立.
2、假设n=k时命题成立,即
1+1/3+1/5+1/(1+2+3+…+k)=2k/(k+1).
则当n=k+1时
左边=1+1/3+1/5+1/(1+2+3+…+k)+1/[1+2+3+…+k+(k+1)]
=[2k/(k+1)]+1/[1+2+3+…+k+(k+1)
=[2k/(k+1)]+2/(k+1)(k+2)
=2(k^2+2k+1)/(k+1)(k+2)
=2(k+1)^2/(k+1)(k+2)
=2(k+1)/(k+2).
∴当n=k+1时命题成立.
综上,命题对所有正整数成立.
证毕.