Multiple choice

The sum of the first $20$ terms of the series $1+\displaystyle\frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+....$, is?

  1. $\displaystyle 38+\frac{1}{2^{20}}$
  2. $\displaystyle 39+\frac{1}{2^{19}}$
  3. $\displaystyle 39+\frac{1}{2^{20}}$
  4. $\displaystyle 38+\frac{1}{2^{19}}$
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D Correct answer
Explanation

The series is 1 + (1 + 1/2) + (1 + 3/4) + (1 + 7/8) + ... = sum_{n=0}^{19} (1 + (2^n - 1)/2^n) = sum (2 - 1/2^n) = 2*20 - sum_{n=0}^{19} (1/2^n). Sum of GP = (1 - (1/2)^20) / (1 - 1/2) = 2(1 - 1/2^20) = 2 - 1/2^19. Total = 40 - (2 - 1/2^19) = 38 + 1/2^19.

AI explanation

The nth term of the series is (2^n - 1) / 2^(n-1), which separates into 2 - 1/2^(n-1). Summing this from 1 to 20 gives the sum of twenty 2s minus the sum of a geometric progression with the first term 1 and common ratio 1/2. This results in 40 minus (1 - (1/2)^20). Simplifying the expression yields 39 + 1/2^19.

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