The sum of an infinite geometric series is 20 | vedclass

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

Question

(B) Let the series be $a, ar, ar^2, \dots$ where $|r| < 1$. Given the sum of the infinite series: $\frac{a}{1 - r} = 20 \quad (i)$. The sum of the squ

Vedclass · Accepted Answer

$3/5$

Vedclass · Answer

(B) Let the series be $a, ar, ar^2, \dots$ where $|r| Given the sum of the infinite series: $\frac{a}{1 - r} = 20 \quad (i)$. The sum of the squares of the terms is $a^2, a^2r^2, a^2r^4, \dots$,which is also an infinite geometric series with first term $a^2$ and common ratio $r^2$. Given the sum of the squares: $\frac{a^2}{1 - r^2} = 100 \quad (ii)$. We can write $(ii)$ as $\frac{a}{1 - r} 	imes \frac{a}{1 + r} = 100$. Substituting $(i)$ into this equation: $20 	imes \frac{a}{1 + r} = 100$,which gives $\frac{a}{1 + r} = 5 \quad (iii)$. Dividing $(i)$ by $(iii)$: $\frac{a/(1 - r)}{a/(1 + r)} = \frac{20}{5} \Rightarrow \frac{1 + r}{1 - r} = 4$. $1 + r = 4 - 4r$ $\Rightarrow 5r = 3$ $\Rightarrow r = 3/5$.

The sum of an infinite geometric series is $20$,and the sum of the squares of its terms is $100$. Find the common ratio of the geometric series.

Similar Questions

If $a, b$ and $c$ form a geometric progression with common ratio $r$,then the sum of the ordinates of the points of intersection of the line $ax + by + c = 0$ and the curve $x + 2y^2 = 0$ is

The first and last terms of a $G.P.$ are $a$ and $l$ respectively; $r$ being its common ratio; then the number of terms in this $G.P.$ is

Which term of the $GP$ $2, 8, 32, \ldots$ is $131072$ (in $^{\text{th}}$)?

If three successive terms of a $G.P.$ with common ratio $r$ $(r > 1)$ are the lengths of the sides of a triangle,and $[r]$ denotes the greatest integer less than or equal to $r$,then $3[r] + [-r]$ is equal to:

The sum of infinite terms of a $G.P.$ is $x$ and on squaring each term of it,the sum becomes $y$. Then the common ratio of this series is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module