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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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

Similar Questions

Let the sum of the first $n$ terms of a non-constant $A.P., a_1, a_2, a_3, \dots$ be $S_n = 50n + \frac{n(n - 7)}{2}A,$ where $A$ is a constant. If $d$ is the common difference of this $A.P.,$ then the ordered pair $(d, a_{50})$ is equal to

If $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}$ is the $A.M.$ of $a$ and $b$,then $n = $

The first term of a $G.P.$ whose second term is $2$ and sum to infinity is $8$,will be

The $4$th term of an $A.P.$ is equal to $3$ times the first term and the seventh term exceeds twice the third term by $1$. Find the first term and the common difference.

The roots of the equation $x^5 - 40x^4 + px^3 + qx^2 + rx + s = 0$ are in $G.P.$ The sum of their reciprocals is $10$. Then the value of $|s|$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module