Multiple choice

The sum of series $\displaystyle {1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ...$ up to n term is-

  1. $n - 1 + {2^{ - n}}$
  2. ${2^{ - n}} - 1$
  3. $n - 1$

  4. None of these
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A Correct answer
Explanation

The series is (1 - 1/2) + (1 - 1/4) + (1 - 1/8) + ... + (1 - 1/2^n). Sum = n - (1/2 + 1/4 + ... + 1/2^n). The sum of the geometric series 1/2 + 1/4 + ... + 1/2^n is (1/2 * (1 - (1/2)^n)) / (1 - 1/2) = 1 - (1/2)^n. Total sum = n - (1 - 1/2^n) = n - 1 + 2^-n.

AI explanation

Using the method of differences, rewrite the nth term of the series as (2^n - 1) / 2^n, which simplifies to 1 - (1/2)^n. Split the series into the sum of n ones minus the sum of a geometric progression with the first term 1/2 and common ratio 1/2. The geometric progression sum is (1/2) * (1 - (1/2)^n) / (1 - 1/2), which equals 1 - 2^-n. Therefore, the total sum is n - (1 - 2^-n), resulting in n - 1 + 2^-n.

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