First slide

Geometric progression

Question

Let $a + {ar}_{1} + {ar}_{1}^{2} + \dots + \infty and a + {ar}_{2} + {ar}_{2}^{2} + \dots + \infty$ be two infinite series of positive numbers with the same first term. The sum of the first series is r₁ and the sum of the second series is r₂. Then the value of (r₁ + r₂) is ______.

Moderate

Solution

$\frac{a}{1 - r_{1}} = r_{1} and \frac{a}{1 - r_{2}} = r_{2}$
Hence, r₁ and r₂ are the roots of $\frac{a}{1 - r} = r$
$\begin{array}{l} \Rightarrow r^{2} - r + a = 0 \\ \Rightarrow r_{1} + r_{2} = 1 \end{array}$

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