Multiple choice

The sum to the first n terms of the series $\dfrac{1}{2} + \dfrac{3}{4} + \dfrac{7}{8} + \dfrac{{15}}{{16}} + {\rm{ }} \ldots $ is $9 + {2^{ - 10}}$. The value of n is

  1. $10$
  2. $9$
  3. $8$

  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) = n - (1/2 + 1/4 + ... + 1/2^n) = n - (1 - 1/2^n). Given sum = 9 + 2^-10. So n - 1 + 2^-n = 9 + 2^-10. Thus n=10.

AI explanation

From the previous derivation, the sum to n terms of the series is given by the formula n - 1 + 2^-n. Equating this expression to the given sum, we have n - 1 + 2^-10 = 9 + 2^-10. Solving this linear equation by adding 1 to both sides yields n = 10.

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