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  4. 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

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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

Sequences and Series

Solution:

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