Multiple choice

The average of (n) numbers is (a). If 2 is added to the first number, 4 is added to the second number, 8 is added to the third number, and each number of this type is added in the same order. Find the new average?

  1. $\(a + \frac{2(2^n - 1)}{n}\)$
  2. $\(a + \frac{2(1 - 2^n)}{n}\)$
  3. $\(n + \frac{2(1 - 2^n)}{n}\)$
  4. $\(a + \frac{2(1 - 2^n)}{a}\)$
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A Correct answer
Explanation

Original sum = na. The numbers added form a geometric progression: 2, 4, 8, 16, ..., 2^n. Sum of this GP = 2(2^n - 1)/(2 - 1) = 2(2^n - 1). New sum = na + 2(2^n - 1). New average = [na + 2(2^n - 1)]/n = a + 2(2^n - 1)/n.

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