If the sum of an infinite G.P. and the sum of | vedclass

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

Question

(B) Let the first series be $a + ar + ar^2 + \dots$ where $|r| < 1$.Then the sum is $S_1 = \frac{a}{1-r} = 3$,which implies $a = 3(1-r)$.The second se

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(B) Let the first series be $a + ar + ar^2 + \dots$ where $|r| Then the sum is $S_1 = \frac{a}{1-r} = 3$,which implies $a = 3(1-r)$.The second series is $a^2 + a^2r^2 + a^2r^4 + \dots$,which is also a $G.P.$ with first term $a^2$ and common ratio $r^2$.The sum is $S_2 = \frac{a^2}{1-r^2} = 3$.Substituting $a = 3(1-r)$ into the second equation:$\frac{[3(1-r)]^2}{1-r^2} = 3$$\frac{9(1-r)^2}{(1-r)(1+r)} = 3$$\frac{3(1-r)}{1+r} = 1$$3 - 3r = 1 + r$$4r = 2$$r = \frac{1}{2}$.

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

Similar Questions

The roots of the equation $x^3-14x^2+56x-64=0$ are in

If $A, B, C$ are the $p^{th}, q^{th},$ and $r^{th}$ terms of a $GP$ respectively,then $A^{q-r} \cdot B^{r-p} \cdot C^{p-q} =$

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

Let $a_1, a_2, a_3, \ldots$ be a $G.P.$ of increasing positive numbers. If $a_3 a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$,then $24(a_1 + a_2 + a_3)$ is equal to

What is the product of $n$ geometric means between $a$ and $b$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module