Question

已知数列｛an｝为公差不为零的等差数列，a1=2，且a2，a4，a8成等比数列。求｛an 3的an

已知数列｛an｝为公差不为零的等差数列，a1=2，且a2，a4，a8成等比数列。求｛an 3的an次方｝的前n项和。。。。请给具体步骤

Answer 1

an=a1+(n-1)d, a1=2
a2，a4，a8成等比数列
a2.a8=(a4)^2
(2+d)(2+7d)=(2+3d)^2
4+16d+7d^2=4+12d+9d^2
d^2-2d=0
d=2
an=2+2(n-1)=2n

let
S=1.3^2+2.3^4+...+n.3^(2n) (1)
9S= 1.3^4+2.3^6+...+n.3^(2n+2) (2)
(2)-(1)
8S =n.3^(2n+2) -(3^2+3^4+...+3^(2n))
=n.3^(2n+2) -(9/8)(3^(2n)-1)
S =8n.3^(2n+2) -9(3^(2n)-1)
= 9 + (72n-9).3^(2n)

bn= an. 3^an
= 2n.3^(2n)
Tn = b1+b2+...+bn
= 2S
=18 + 18(8n-1).3^(2n)