Series

1. Understand the problem:

We are given the expansion of and need to evaluate the sum:

where , the binomial coefficients.

2. Express the general term:

Each term in the sum is of the form . We can simplify this ratio:

This comes from the property of binomial coefficients: .

3. Simplify the general term:

Substituting the simplified ratio into the general term:

4. Rewrite the sum:

The original sum now becomes:

5. Evaluate the sum:

This is an arithmetic series. Let , then the sum is:

Substituting back :

6. Match the result to the options:

The simplified form corresponds to option (A).

Correct Answer: (A)