Question

Find the sum to infinity of the following arithmetico - geometric sequence:

`1, -4/3, 7/9, -10/27 ...`

Answer 1

The given sequence can be written as :

`1 xx 1, 4 xx (-1/3), 7 xx (-1/3)^2, 10 xx (-1/3)^3, ...`

This is arithmetico-geometric progression with

a = 1, d = 3, r = `-1/3`, where |r| = `|-1/3| = 1/3 < 1`

∴ sum to infinity of the A.G.P. is given by

S = `"a"/(1 - "r") + "dr"/(1 - "r")^2`

= `1/(1 - (-1/3)) + (3(-1/3))/[1 - (-1/3)]^2`

= `1/((4/3)) - 1/((16/9))`

= `3/4 - 9/16`

= `3/16`