The sum of an infinite geometric series is 3. | vedclass

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

Question

(A) Let the first term of the infinite geometric series be $a$ and the common ratio be $r$. The sum of an infinite geometric series is given by $S = \

Vedclass · Accepted Answer

$\frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, .....$

Vedclass · Answer

(A) Let the first term of the infinite geometric series be $a$ and the common ratio be $r$. The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$.Given,$\frac{a}{1-r} = 3$,so $a = 3(1-r)$ ..... $(i)$.The series formed by the squares of its terms is $a^2, a^2r^2, a^2r^4, .....$. This is also an infinite geometric series with first term $a^2$ and common ratio $r^2$.The sum of this series is $\frac{a^2}{1-r^2} = 3$ ..... $(ii)$.Substituting $a = 3(1-r)$ into $(ii)$:$\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 \implies r = \frac{1}{2}$.Substituting $r = \frac{1}{2}$ into $(i)$:$a = 3(1 - \frac{1}{2}) = 3(\frac{1}{2}) = \frac{3}{2}$.The series is $a, ar, ar^2, .....$,which is $\frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, .....$.

The sum of an infinite geometric series is $3$. $A$ series,which is formed by squares of its terms,has the sum also $3$. The first series is

Similar Questions

If the ${n^{th}}$ terms of two arithmetic progressions $(A.P.)$ are $3n + 8$ and $7n + 15$,then the ratio of their ${12^{th}}$ terms will be:

If the sum of $n$ terms of a $G.P.$ is $255$,the $n^{th}$ term is $128$,and the common ratio is $2$,then the first term will be:

If in an infinite $G.P.$ the first term is equal to twice the sum of the remaining terms,then its common ratio is

The $n^{th}$ term of the series $1 + \frac{4}{5} + \frac{7}{5^2} + \frac{10}{5^3} + \dots$ will be

If $a_1, a_2, a_3, . . . , a_n, . . .$ are in $A.P.$ such that $a_4 - a_7 + a_{10} = m$,then the sum of the first $13$ terms of this $A.P.$ is .............. $m$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module