Final answer:
The question involves finding the new mean after increasing each of n items by powers of 3. The increase follows a geometric progression, and the new mean is found by adding the mean value of this progression to the original mean x, leading to option B as the correct answer.
Step-by-step explanation:
The student's question involves determining the new mean when each of n items, initially with a mean of x, is increased successively by powers of 3 (i.e., 3, 3², 3³, ..., 3ⁿ). To find the new mean, we must add the average increase to the original mean. The increase for each term is the sum of a geometric series, with the first term being 3 and the common ratio also being 3. The sum of this series is:
S = 3 * (1 - 3^n) / (1 - 3)
= (3/2) * (3^n - 1)
The average of the increases is the total sum divided by n, which is (3/2n) * (3^n - 1).
Therefore, the new mean is the original mean (x) plus this average increase:
New mean = x + (3/2n) * (3^n - 1)
This matches option B in the student's question. Thus, the correct answer is:
B. x + 3*(3^n - 1)/(2n)