Question

The mean of \(n\) items is \(\bar{x}\). If each item is successively increased by \(3, 3^2, 3^3, \ldots, 3^n\), then the new mean will be:

• A. \(\bar{x} + \dfrac{3^{n+1}}{2n}\)
• B. \(\bar{x} + \dfrac{3(3^n-1)}{3n}\)
• C. \(\bar{x} + \dfrac{3^n}{3n}\)
• D. \(\bar{x} + \dfrac{3(3^n-1)}{2n}\)

Hint:

When each observation is increased by different amounts, the new mean equals: [ textOld mean + fractextSum of increasesn ]

Answer 1

The Correct Option is D

Step 1: Recall the relation between mean and total sum. If the mean of \(n\) items is \(\bar{x}\), then the sum of the items is: \[ \text{Sum} = n\bar{x} \] Step 2: Find the total increment added to all items. Each item is increased by: \[ 3, 3^2, 3^3, \ldots, 3^n \] This is a geometric progression with: \[ a = 3,\quad r = 3 \] Sum of this G.P.: \[ S = \frac{a(r^n-1)}{r-1} = \frac{3(3^n-1)}{2} \] Step 3: Find the increase in mean. Increase in mean: \[ = \frac{\text{Total increase}}{n} = \frac{3(3^n-1)}{2n} \] Step 4: Write the new mean. \[ \text{New mean} = \bar{x} + \frac{3(3^n-1)}{2n} \] Step 5: Final conclusion. The correct option is \(\boxed{(D)}\).