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