Multiple choice

The sum to $50$ terms of the series $\dfrac {1}{2} + \dfrac {3}{4} + \dfrac {7}{8} + \dfrac {15}{16} + ....$ is equal to

  1. $2^{50} - 51$
  2. $1 - 2^{-50}$
  3. $2^{-50}+49$
  4. $2^{50} - 1$
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C Correct answer
Explanation

The series is 1/2 + 3/4 + 7/8 + 15/16 + ... which can be written as (1 - 1/2) + (1 - 1/4) + (1 - 1/8) + (1 - 1/16) + ... to 50 terms. This equals 50 - (1/2 + 1/4 + 1/8 + ... + 1/2^50). The sum of the geometric series is (1/2 * (1 - (1/2)^50)) / (1 - 1/2) = 1 - 1/2^50. Total sum = 50 - (1 - 1/2^50) = 49 + 1/2^50.

AI explanation

The general term of the series is (2^n - 1) / 2^n, which can be split into 1 - 1/(2^n). The sum to 50 terms is the sum of fifty 1s minus the sum of the geometric series 1/2 + 1/4 + 1/8 to 50 terms. The sum of this infinite-like geometric progression is a/(1-r), giving (1/2) / (1 - 1/2) = 1, but calculating exactly to 50 terms using Sn = a(1-r^n)/(1-r) gives a sum of 1 - 1/(2^50). Subtracting this from 50 gives 50 - (1 - 2^-50) = 49 + 2^-50.

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