Question

Find the sum of G.P. :

`(x + y)/(x - y) + 1 + (x - y)/(x + y) + ..........` upto n terms. 

Answer 1

Given G.P. : `(x + y)/(x - y) + 1 + (x - y)/(x + y) + ..........` upto n terms
Here,

First term, a =`(x + y)/(x - y)`

Common ratio, r = `1/((x + y)/(x - y)) = (x - y)/(x + y)`  ...(∵ r < 1)

Number of terms to be added = n

∴ `S_n = (a(1 - r^n))/(1 - r)`

`=> S_n = ((x + y)/(x - y)(1 - ((x - y)/(x + y))^n))/(1 - ((x - y)/(x + y)))`

= `((x + y)/(x - y)(1 - ((x - y)/(x + y))^n))/((x + y - x + y)/(x + y))`

= `((x + y)/(x - y)(1 - ((x - y)/(x + y))^n))/((2y)/(x + y))`

= `((x + y)^2(1 - ((x - y)/(x + y))^n))/(2y(x - y))`