Question

求前n项和:(a+1/a)^2 ,(a^2+1/a^2)^2 ,(a^3+1/a^3)^2

Answer 1

(a+1/a)^2=a^2+2+1/a^2

(a^2+1/a^2)^2=a^4+2+1/a^4

(a^3+1/a^3)^2=a^6+2+1/a^6

.

Sn=a^2+2+1/a^2+a^4+2+1/a^4+a^6+2+1/a^6+.+a^2n+2+1/a^2n

Sn=(a^2+a^4+a^6+.+a^2n)+(1/a^2+1/a^4+1/a^6+.+1/a^2n)+ 2n

当a^2=1时

Sn=n+n+2n=4n

当a^2≠1时

a^2+a^4+a^6+.+a^2n

=a^2(1-a^2n)/(1-a^2)

1/a^2+1/a^4+1/a^6+.+1/a^2n

=(1/a^2)*(1-1/a^2n)/(1-1/a^2)

=(1-1/a^2n)/(a^2-1)

∴Sn=a^2(1-a^2n)/(1-a^2)+(1-1/a^2n)/(a^2-1)+2n

=[a^(2n+2)-a^2+1-(1/a^2n)]/(a^2-1)+2n