数列a（n）满足a（n+1）+（-1）^na（n）=2n-1求a（n）的前60项和（填空题）这道题答案用的分奇偶情况讨论写的老长这要按平常做填空题时间根本不可能做完所以我想请教一下有没有简便办法

Answer 1

因有(-1)^n,所以只能分奇偶讨论.
但确实可以短一点.
a(2n-1+1) + (-1)^(2n-1)a(2n-1) = 2(2n-1)-1 = 4n-3 = a(2n) - a(2n-1),
a(2n+1) + (-1)^(2n)a(2n) = 2(2n)-1 = 4n-1 = a(2n+1) + a(2n),
a(2n+1)+a(2n-1) = [a(2n+1)+a(2n)]-[a(2n) -a(2n-1)] = (4n-1)-(4n-3) = 2.
a(1)+a(3)+...+a(2*29-1) + a(2*29+1) = 2*29 = 58.
a(2n+1+1)+(-1)^(2n+1)a(2n+1) = 2(2n+1)-1 = 4n+1 = a(2n+2)-a(2n+1).
a(2n+2)+a(2n) = [a(2n+2)-a(2n+1)]+[a(2n+1)+a(2n)] = (4n+1) + (4n-1) = 8n.
a(2)+a(4)+...+a(2*29) + a(2*29+2) = 8(1+2+...+29) = 4*29*30=3480
a(1)+a(2)+...+a(60) = 58+3480 = 3538