已知数列｛an｝中,an=1+1/2+1/3+...+1/n,记sn=a1+a2+...+an用数学归纳法证明sn=(n+1)an-n

题目

已知数列｛an｝中,an=1+1/2+1/3+...+1/n,记sn=a1+a2+...+an用数学归纳法证明sn=(n+1)an-n

答案

(1) 当n=1时
S1=(1+1)a1-1
=2a1-1
=2-1
=1
∴当n=1时sn=(n+1)an-n成立
(2)假设n=k时sn=(n+1)an-n成立
即有 Sk=(k+1)ak-k
Sk+1=1+1/2+1/3+...+1/k+1/(k+1)
=Sk+1/(k+1)
=(k+1)ak-k+1/(k+1)
=(k+1)/k-k+1/(k+1)
=1+1/k-k+1/(k+1)
=(k+2)/(k+1)-(k+1)
=(k+2)ak+1-(k+1)
即当n=k+1时 Sk+1=(k+2)ak+1-(k+1) 成立
综上所述,当n∈N* 时有sn=(n+1)an-n 成立