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Sum of Infinite Geometric Series

CMAT 2021 · Quantitative Techniques & Data Interpretation · Medium · Progressions

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Find the sum of the following series (with infinite terms): 22,43,423,…2\sqrt{2}, \frac{4}{\sqrt{3}}, \frac{4\sqrt{2}}{3}, \dots

  1. A.

    23(3+2)2\sqrt{3}(\sqrt{3} + \sqrt{2})

  2. B.

    23(3−2)2\sqrt{3}(\sqrt{3} - \sqrt{2})

  3. C.

    26(3+2)2\sqrt{6}(\sqrt{3} + \sqrt{2})

  4. D.

    26(3−2)2\sqrt{6}(\sqrt{3} - \sqrt{2})

Answer

C

Explanation

First term a=22a = 2\sqrt{2}. Common ratio r=4/322=23r = \frac{4/\sqrt{3}}{2\sqrt{2}} = \sqrt{\frac{2}{3}}. Sum S=a1−r=221−23=263−2S = \frac{a}{1 - r} = \frac{2\sqrt{2}}{1 - \sqrt{\frac{2}{3}}} = \frac{2\sqrt{6}}{\sqrt{3} - \sqrt{2}}. Rationalizing denominator: S=26(3+2)S = 2\sqrt{6}(\sqrt{3} + \sqrt{2}).

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