The sum of an infinite geometric series is 20 | vedclass

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(B) Let the series be $a, ar, ar^2, \dots$ where $|r| < 1$. Given the sum of the infinite series: $\frac{a}{1 - r} = 20 \quad (i)$. The sum of the squ

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$3/5$

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(B) Let the series be $a, ar, ar^2, \dots$ where $|r| Given the sum of the infinite series: $\frac{a}{1 - r} = 20 \quad (i)$. The sum of the squares of the terms is $a^2, a^2r^2, a^2r^4, \dots$,which is also an infinite geometric series with first term $a^2$ and common ratio $r^2$. Given the sum of the squares: $\frac{a^2}{1 - r^2} = 100 \quad (ii)$. We can write $(ii)$ as $\frac{a}{1 - r} 	imes \frac{a}{1 + r} = 100$. Substituting $(i)$ into this equation: $20 	imes \frac{a}{1 + r} = 100$,which gives $\frac{a}{1 + r} = 5 \quad (iii)$. Dividing $(i)$ by $(iii)$: $\frac{a/(1 - r)}{a/(1 + r)} = \frac{20}{5} \Rightarrow \frac{1 + r}{1 - r} = 4$. $1 + r = 4 - 4r$ $\Rightarrow 5r = 3$ $\Rightarrow r = 3/5$.

The sum of an infinite geometric series is $20$,and the sum of the squares of its terms is $100$. Find the common ratio of the geometric series.

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