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प्रश्न
Find the sum of G.P. :
`(x + y)/(x - y) + 1 + (x - y)/(x + y) + ..........` upto n terms.
उत्तर
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))`
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