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Geometric Progression: Sum Ratio

XAT 2016 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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Consider the set of numbers {1,3,32,33,…,3100}\{1, 3, 3^2, 3^3, \dots, 3^{100}\}. The ratio of the last number and the sum of the remaining numbers is closest to:

  1. A.

    1

  2. B.

    2

  3. C.

    3

  4. D.

    50

  5. E.

    99

Answer

B

Explanation

Last number = 31003^{100}. Sum of remaining numbers = 1+3+32+⋯+399=1(3100−1)3−1=3100−121 + 3 + 3^2 + \dots + 3^{99} = \frac{1(3^{100} - 1)}{3 - 1} = \frac{3^{100} - 1}{2}. Ratio = 31003100−12=2⋅31003100−1≈2\frac{3^{100}}{\frac{3^{100} - 1}{2}} = \frac{2 \cdot 3^{100}}{3^{100} - 1} \approx 2.

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