Let {a_n} be the {n^{th}} term of a G.P. of p | 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\limits_{n = 1}^{100} {{a_{2n}}} = a_2 + a_4 + \dots + a_{2

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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 $\alpha = \sum\limits_{n = 1}^{100} {{a_{2n}}} = a_2 + a_4 + \dots + a_{200}$.This is a $G$.$P$. with first term $ar$ and common ratio $r^2$ having $100$ terms.So,$\alpha = ar(1 + r^2 + r^4 + \dots + r^{198})$.Given $\beta = \sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = a_1 + a_3 + \dots + a_{199}$.This is a $G$.$P$. with first term $a$ and common ratio $r^2$ having $100$ terms.So,$\beta = a(1 + r^2 + r^4 + \dots + r^{198})$.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 a $G$.$P$. 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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