Question

The sum of the first two terms of a GP is 5/3 and the sum to infinity of the series is 3. The common ratio is
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Answer 1

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Answer 2

let the first term of the GP be 'a' with common ratio 'r'

given

The sum of the first two terms is 5/3

and the sum to infinity of the series is 3

$a + ar = \frac{5}{3} \\ \\ a(1 + r) = \frac{5}{3} \\ \\ \\ also \\ \\ \frac{a}{1 - r} = 3 \\ \\ but \\ \\ a = \frac{5}{3(1 + r)} \\ \\ \frac{5}{3(1 - {r}^{2}) } = 3 \\ \\ \frac{5}{9} = 1 - {r}^{2} \\ \\ {r}^{2} = \frac{4}{9} \\ \\ r = \frac{2}{3} \\ \\ r = - \frac{2}{3} \\ \\ if \: \: r = \frac{2}{3} \\ \\ a = 1 \\ \\ if \: \: r = - \frac{2}{3} \\ \\ a = 5 \\ \\ also \: r < 1$

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