If the ratio of the sum of first three terms and the sum of first six terms of

Discussion Forum : Sequences And Series

Question -

If the ratio of the sum of first three terms and the sum of first six terms of a G.P. be 125 : 152, then the common ratio $r$ is ___.

Options:

A .

\frac{3}{5}

B .

\frac{5}{3}

C .

\frac{2}{3}

D .

\frac{3}{2}

Answer: Option A
:
A

Given that $\frac{a + a r + a r^{2}}{a + a r + a r^{2} + a r^{3} + a r^{4} + a r^{5}} = \frac{125}{152} .$

$\Rightarrow \frac{1 + r + r^{2}}{1 + r + r^{2} + r^{3} + r^{4} + r^{5}} = \frac{125}{152} i.e., \frac{1 + r + r^{2}}{(1 + r + r^{2}) + r^{3} (1 + r + r^{2})} = \frac{125}{152} \Rightarrow \frac{1}{1 + r^{3}} = \frac{125}{152} \Rightarrow r^{3} = \frac{152}{125} - 1 = \frac{27}{125} \Rightarrow r = \frac{3}{5}$

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