Multiple choice

Mean of 'n' items is $\bar{x}$. If these n items are successively increased by $2, 2^2, 2^3$,.... $2^n$, then the new mean is?

  1. $\bar{x}+\dfrac{2^{n+1}}{n}$
  2. $\bar{x}+\dfrac{2^{n+1}}{n}-\dfrac{2}{n}$
  3. $\bar{x}+\dfrac{2^n}{n}$
  4. $\bar{x}+2^n$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The original sum of n items is n * x_bar. The new sum is (n * x_bar) + (2 + 2^2 + ... + 2^n). The sum of the geometric series 2 + 2^2 + ... + 2^n is 2 * (2^n - 1) / (2 - 1) = 2^(n+1) - 2. The new mean is (n * x_bar + 2^(n+1) - 2) / n = x_bar + (2^(n+1) - 2) / n, which simplifies to x_bar + 2^(n+1)/n - 2/n.

AI explanation

The new mean is calculated as the sum of the original items plus the sum of the added geometric progression, all divided by n. The sum of the original n items is n times x-bar, and the sum of the geometric series 2, 2^2, ..., 2^n is 2^(n+1) - 2. Combining these, the new sum is n*x-bar + 2^(n+1) - 2, so dividing by n gives the new mean as x-bar + 2^(n+1)/n - 2/n.

Keep practicing — related questions