已知数列an的通项an=(2n-1)·3^n-1,求an的前n项和Sn
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已知数列an的通项an=(2n-1)·3^n-1,求an的前n项和Sn
已知数列an的通项an=(2n-1)·3^n-1,求an的前n项和Sn
已知数列an的通项an=(2n-1)·3^n-1,求an的前n项和Sn
an=(2n-1)·3^(n-1)
= 2[ n.3^(n-1) ] - 3^(n-1)
Sn = a1+a2+...+an
= 2[∑(i:1->n) i.3^(i-1) ] - (3^n-1)/2
let
S = 1.3^0+ 2.3^1+.+n.3^(n-1) (1)
3S = 1.3^1+ 2.3^2+.+n.3^n (2)
(2)-(1)
2S = n.3^n - [ 3^0+3^1+.+3^(n-1) ]
=n.2^n - (3^n-1)/2
S = 2n.2^n - (3^n-1)
Sn = 2[∑(i:1->n) i.3^(i-1) ] - (3^n-1)/2
= 2S -(3^n-1)/2
= 4n.2^n - (3/2)( 3^n-1)