If the first term of an infinite geometric se | vedclass

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

Question

(B) Let the infinite geometric series be $a, ar, ar^2, ar^3, \dots$ where $a = 1$.According to the problem,each term is equal to the sum of all subseq

Vedclass · Accepted Answer

$1/8$

Vedclass · Answer

(B) Let the infinite geometric series be $a, ar, ar^2, ar^3, \dots$ where $a = 1$.According to the problem,each term is equal to the sum of all subsequent terms.For the first term: $a = ar + ar^2 + ar^3 + \dots$Since $a = 1$,we have $1 = ar + ar^2 + ar^3 + \dots$This is an infinite geometric series with first term $ar$ and common ratio $r$.The sum of an infinite geometric series is given by $S = \frac{	ext{first term}}{1 - r}$.So,$1 = \frac{ar}{1 - r}$.Substituting $a = 1$: $1 = \frac{r}{1 - r}$.$1 - r = r \implies 2r = 1 \implies r = 1/2$.The fourth term is $T_4 = ar^3$.$T_4 = 1 	imes (1/2)^3 = 1/8$.

If the first term of an infinite geometric series is $1$ and each term is equal to the sum of all the terms that follow it,then what is its fourth term?

Similar Questions

Insert two numbers between $3$ and $81$ so that the resulting sequence is a $G.P.$

If $a, b$ and $c$ are three distinct numbers in $G.P.$ and $a + b + c = xb$,then $x$ cannot be:

For a $G.P.$,if $(m+n)^{\text{th}}$ term is $p$ and $(m-n)^{\text{th}}$ term is $q$,then the $m^{\text{th}}$ term is $.........$

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

Divide $155$ into three parts such that the three numbers are in a Geometric Progression $(GP)$ and the first term is $120$ less than the third term.

Vedclass Test Series

Exam Paper Generator

Online Exam Module