Arithmetic Progression

In an arithmetic progression (A.P.) with common difference , the terms can be written as:

The mean of these terms is:

The mean of their squares is:

Now, the difference between the mean of squares and the square of the mean is:

This expression represents the variance of the A.P. terms. Using known formulas:

After simplifying, we get:

Difference =

So, the correct answer is Option (C):

We have an arithmetic progression (A.P.) involving the given expressions.

An arithmetic progression means the difference between consecutive terms is constant.

Given terms:

First term:

Second term:

Third term:

Let’s find the differences:

Second - First:

Third - Second:

Since these are in A.P., the differences must be equal:

Bring all terms to one side:

Convert to base 3:

So the equation becomes:

Now simplify:

So:

Let

Now exponentiate both sides (base 3):

This is a transcendental equation. By trying values, we find that:

At , both sides are equal.

So, the correct answer is Option (B):