Question

一道关于数学归纳法的题目用数学归纳法证明：1/(1*2) + 1/(2*3) + ...+ 1/n(n+1) = n/n

Answer 1

证明：当n=1时..左边=1/(1*2)..右边=1/(1*2)..左边=右边..显然成立..

假设当n=k时..则1/(1*2) + 1/(2*3) + ...+ 1/k(k+1)=k/(k+1)成立...

当n=k+1时..

1/(1*2) + 1/(2*3) + ...+ 1/k(k+1)+ 1/(k+1)(k+2)=k/(k+1)+1/(k+1)(k+2)=(k+1)^2/(k+1)(k+2)=(k+1)/(k+2)=(k+1)/(k+1+1)

所以n=k+1时成立...

综上所述..

1/(1*2) + 1/(2*3) + ...+ 1/n(n+1) = n/n+1