Multiple choice

The sum of the first n terms of the series $$1⁄2 + 3⁄4 + 7⁄8 + 15⁄16 + …$$ is equal to

  1. 2n - n + 1

  2. 1 - 2 - n

  3. n + 2-n - 1

  4. 2n - 1
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C Correct answer
AI explanation

Each term of the series can be written as the fraction (2^k - 1) divided by 2^k, which simplifies to 1 - (1 divided by 2^k). By splitting the terms, the sum of the first n terms becomes the sum of n ones minus the sum of the geometric progression 1/2, 1/4, and so on up to 1/(2^n). The sum of this geometric progression is 1 multiplied by (1 - (1/2)^n) divided by (1 - 1/2), which equals 1 - 2^(-n). Subtracting this from n gives the final result of n + 2^(-n) - 1.

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