Let {a_n} be the {n^{th}} term of the G.P. of | vedclass

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(A) Let the $G$.$P$. be $a, ar, ar^2, \dots$ where $a > 0$ and $r > 0$.Given $\alpha = \sum_{n=1}^{100} a_{2n} = a_2 + a_4 + \dots + a_{200}$.This is

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$\frac{\alpha }{\beta }$

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(A) Let the $G$.$P$. be $a, ar, ar^2, \dots$ where $a > 0$ and $r > 0$.Given $\alpha = \sum_{n=1}^{100} a_{2n} = a_2 + a_4 + \dots + a_{200}$.This is a $G$.$P$. with first term $a_2 = ar$ and common ratio $r^2$.So,$\alpha = ar + ar^3 + \dots + ar^{199} = ar(1 + r^2 + r^4 + \dots + r^{198})$.Given $\beta = \sum_{n=1}^{100} a_{2n-1} = a_1 + a_3 + \dots + a_{199}$.This is a $G$.$P$. with first term $a_1 = a$ and common ratio $r^2$.So,$\beta = a + ar^2 + \dots + ar^{198} = a(1 + r^2 + r^4 + \dots + r^{198})$.Dividing $\alpha$ by $\beta$:$\frac{\alpha}{\beta} = \frac{ar(1 + r^2 + r^4 + \dots + r^{198})}{a(1 + r^2 + r^4 + \dots + r^{198})} = r$.Thus,the common ratio is $\frac{\alpha}{\beta}$.

Let ${a_n}$ be the ${n^{th}}$ term of the $G$.$P$. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $,such that $\alpha \ne \beta $,then the common ratio is

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