Question

数列求和题目

已知一个数列的首项为15，第二项开始依次为81、 13×27、17×81、21×243、25×729。请求出这个数列前n项的和（n取正整数）。

Answer 1

a1=15 =5x3^1
a2=81= (5+4)x3^2
a3=13×27=(5+2x4)x3^3
a4=17×81 =(5+3x4)x3^4
an
= [5+4(n-1) ] . 3^n
=(4n+1).3^n
= 4n.3^n + 3^n
=bn +cn
bn= 4n.3^n
cn = 3^n
Tn= c1+c2+...+cn = (3/2)( 3^n-1)
Pn= b1+b2+...+bn
=4x( 1x3^1 + 2x3^2+....+n.3^n) (1)
3(1)
3Pn =4x( 1x3^2 + 2x3^3+....+n.3^(n+1) ) (2)
(2)-(1)
2Pn =4n.3^(n+1) - 4( 3^1+3^2+...+3^n )
=4n.3^(n+1) - 6( 3^n-1)
Pn =2n.3^(n+1) - 3( 3^n-1)
Sn
=a1+a2+...+an
=Pn +Tn
=2n.3^(n+1) - 3( 3^n-1) +(3/2)( 3^n-1)
=2n.3^(n+1) - (3/2)( 3^n-1)