The sum of a certain infinite geometric series is 20 . When all the terms in the series are squared, the sum of the resulting series is 80 . If the first term of the original series is expressed in lowest terms as (p/q),(p, q ∈ N) then the value of (p+q) is

Question

The sum of a certain infinite geometric series is 20 . When all the terms in the series are squared, the sum of the resulting series is 80 . If the first term of the original series is expressed in lowest terms as (p/q),(p, q ∈ N) then the value of (p+q) is (A) 21 (B) 22 (C) 23 (D) 24

Tardigrade Admin · Accepted Answer

S =( a /1- r )=20 ⇒ ( a 2/1- r 2)=80 ⇒ r =(2/3) ∴ a=(20/3)=(p/q) ⇒(p+q)=23

Q. The sum of a certain infinite geometric series is 20 . When all the terms in the series are squared, the sum of the resulting series is 80 . If the first term of the original series is expressed in lowest terms as $\frac{p}{q}, (p, q \in N)$ then the value of $(p + q)$ is

21

22

23

24

$S = \frac{a}{1 - r} = 20 \Rightarrow \frac{a ^{2}}{1 - r ^{2}} = 80 \Rightarrow r = \frac{2}{3}$
$∴ a = \frac{20}{3} = \frac{p}{q} \Rightarrow (p + q) = 23$

Q. The sum of a certain infinite geometric series is 20 . When all the terms in the series are squared, the sum of the resulting series is 80 . If the first term of the original series is expressed in lowest terms as qp​,(p,q∈N) then the value of (p+q) is

21

22

23

24

S=1−ra​=20⇒1−r2a2​=80⇒r=32​ ∴a=320​=qp​⇒(p+q)=23

Q. The sum of a certain infinite geometric series is 20 . When all the terms in the series are squared, the sum of the resulting series is 80 . If the first term of the original series is expressed in lowest terms as $\frac{p}{q}, (p, q \in N)$ then the value of $(p + q)$ is

$S = \frac{a}{1 - r} = 20 \Rightarrow \frac{a ^{2}}{1 - r ^{2}} = 80 \Rightarrow r = \frac{2}{3}$
$∴ a = \frac{20}{3} = \frac{p}{q} \Rightarrow (p + q) = 23$