The sum of the first two terms of a geometric progression is x and sum…

Explanation

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 = S_n - S_{n - 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 = xr^{n - 2}

= yr^{n - 2}

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