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The sum of the first two terms of a geometric progression is...

The sum of the first two terms of a geometric progression is x and sum of the last terms is y. If there are n terms in all, then the common ratio is

A.

\(\frac{x}{y}\)

B.

\(\frac{y}{x}\)

C.

(\(\frac{x}{y}\))\(\frac{1}{n - 2}\)

D.

(\(\frac{y}{x}\))\(\frac{1}{n - 2}\)

Correct answer is D

Sum of nth term of a G.P = Sn = \(\frac{ar^n - 1}{r - 1}\)

sum of the first two terms = \(\frac{ar^2 - 1}{r - 1}\)

x = a(r + 1)

sum of the last two terms = Sn - Sn - 2

= \(\frac{ar^n - 1}{r - 1}\) - \(\frac{(ar^{n - 1})}{r - 1}\)

= \(\frac{a(r^n - 1 - r^{n - 2} + 1)}{r - 1}\) (r2 - 1)

∴ \(\frac{ar^{n - 2}(r + 1)(r - 1)}{1}\)= arn - 2(r + 1) = y

= a(r + 1)r^n - 2

y = xrn - 2

= yrn - 2

\(\frac{y}{x}\) = r = (\(\frac{y}{x}\))\(\frac{1}{n - 2}\)