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 be $a$ and the common ratio be $r$. The sum of the infinite geometric series is given by $S = \frac{a}{1-r} = 3$,so $a = 3(1-r)

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Let the first term be $a$ and the common ratio be $r$. The sum of the infinite geometric series is given by $S = \frac{a}{1-r} = 3$,so $a = 3(1-r) \dots (i)$.The series formed by squaring its terms is $a^2, a^2r^2, a^2r^4, \dots$,which is a geometric series with first term $a^2$ and common ratio $r^2$. Its sum is $\frac{a^2}{1-r^2} = 3$.Substituting $a = 3(1-r)$ into the second equation: $\frac{9(1-r)^2}{1-r^2} = 3$.Since $1-r^2 = (1-r)(1+r)$,we have $\frac{9(1-r)^2}{(1-r)(1+r)} = 3$,which simplifies to $\frac{3(1-r)}{1+r} = 1$.Solving for $r$: $3 - 3r = 1 + r$,so $4r = 2$,which gives $r = \frac{1}{2}$.Substituting $r = \frac{1}{2}$ into $(i)$: $a = 3(1 - \frac{1}{2}) = \frac{3}{2}$.The series is $\frac{3}{2}, \frac{3}{2}(\frac{1}{2}), \frac{3}{2}(\frac{1}{2})^2, \dots$,which is $\frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}, \dots$.

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

Similar Questions

Find the sum of the series $1 + 3 + 7 + 15 + 31 + \dots$ up to $n$ terms.

If the sum of the first $11$ terms of the series ${\left( {1\frac{4}{7}} \right)^2} + {\left( {1\frac{5}{7}} \right)^2} + {\left( {1\frac{6}{7}} \right)^2} + {2^2} + {\left( {2\frac{1}{7}} \right)^2} + \dots$ is $\frac{11}{7}\lambda$,then $\lambda$ is equal to:

If $|x| < 1$,what is the sum of the series $1 + 2x + 3x^2 + 4x^3 + \dots \infty$?

For an odd integer $n \ge 1$,the value of $n^3 - (n-1)^3 + (n-2)^3 - (n-3)^3 + \dots + (-1)^{n-1} 1^3$ is:

The sum to infinity of the progression $9 - 3 + 1 - \frac{1}{3} + \dots$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module