用数学归纳法证明1∕1×2+1∕2×3+1∕3×4+.+1∕n(n+1)=n∕n+1

Question

用数学归纳法证明1∕ 1×2+1∕ 2×3+1∕ 3×4+.+1∕n(n+1)=n∕n+1

Answer 1

当n=1时1/1x2=1/2=1/(1+1)成立
设n=k时1∕ 1×2+1∕ 2×3+1∕ 3×4+.+1∕k(k+1)=k∕(k+1)成立
则n=k+1时
1∕ 1×2+1∕ 2×3+1∕ 3×4+.+1∕k(k+1)+1/(k+1)(k+2)
=k/(k+1)+1/(k+1)(k+2)
=k/(k+1)+1/(k+1)-1/(k+2)
=1-1/(k+2)=(k+1)/(k+2)得证
所以1∕ 1×2+1∕ 2×3+1∕ 3×4+.+1∕n(n+1)=n∕n+1成立