高中数学,已知数列{f（n）}满足f（n+1）+f(n)×(-1)^n=2n-1,求此数列前60项和.

题目

高中数学,已知数列{f（n）}满足f（n+1）+f(n)×(-1)^n=2n-1,求此数列前60项和.

答案

The answer is 1830.
Let a=f(1).By induction,one may easily prove that for any n>=0,
f(4n+1)=a,
f(4n+2)=8n+1+a,
f(4n+3)=2-a,
f(4n+4)=8n+7-a.
Therefore,
f(1)+f(2)+...+f(60)
=sum_{n=0}^{14}(f(4n+1)+f(4n+2)+f(4n+3)+f(4n+4))
=sum_{n=0}^{14}(16n+10)
=16*14*15/2+150
=1830.