Multiple choice

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

  1. $n - 1 - 2^{-n}$
  2. $1$
  3. $n - 1 + 2^{-n}$
  4. $1 + 2^{-n}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The series is 1/2 + 3/4 + 7/8 + ... + (2^n - 1)/2^n. This can be rewritten as (1 - 1/2) + (1 - 1/4) + (1 - 1/8) + ... + (1 - 1/2^n). Summing these n terms gives n - (1/2 + 1/4 + ... + 1/2^n), which simplifies to n - (1 - 2^-n) = n - 1 + 2^-n.

AI explanation

The nth term of the given series can be written as the fraction whose numerator is 2 to the power of n minus 1 and whose denominator is 2 to the power of n. By splitting the fraction, the term becomes 1 minus 2 to the power of negative n. Adding the first n terms gives a sum of n minus the geometric series created by the fractional parts, which equals n minus the quantity 1 minus 2 to the power of negative n. Therefore, the final sum is n minus 1 plus 2 to the power of negative n.

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