Question

Find S_n of the following arithmetico - geometric sequence: 

1, 4x, 7x², 10x³, 13x⁴, …

Answer 1

1, 4x, 7x², 10x³, 13x⁴, …

Here, 1, 4, 7, 10, 13, … are in A.P.

∴ a = 1, d = 3

∴ n^th term = a + (n – 1)d

= 1 + (n – 1)(3)

= 3n – 2

1, x, x², x³, … are in G.P.

∴ a = 1, r = x,

∴ n^th term = ar^n–1 = x^n–1

∴ n^th term of arithmetico-geometric sequence is

t_n = (3n – 2)x^n–1

∴ S_n = 1 + 4x + 7x² + 10x³ + …+ (3n – 2)x^n–1  ...(i)

Multiplying throughout by x, we get 

xS_n = x + 4x² + 7x³ + 10x⁴ + …+ (3n – 2)xⁿ  ...(ii)

Equation (i) – equation (ii), we get

S_n – xS_n = 1+ 3x + 3x² + 3x³ + … + 3x^n–1 – (3n – 2).xⁿ

∴ (1 – x)S_n = `1 + 3x ((1 - x^("n" -1))/(1 - x)) - (3"n" - 2).x^"n"`

∴ (1 – x)S_n = `1 - (3"n" - 2).x^"n" + (3x(1 - x^("n" -1)))/(1 - x)`

∴ S_n = `(1 - (3"n" - 2)x^"n")/(1 - x) + (3x(1 -x^("n" -1)))/(1 - x)^2`