Binomial Theorem

Step 1: Let the A.P. have first term and common difference .
Step 2: Since these three terms are in G.P., the square of the middle term equals the product of the other two:
Step 3: Expand both sides:
Step 4: Simplify:
Step 5: Since the A.P. is non-constant, , hence:
Step 6: Substitute in the terms:
Step 7: The common ratio of the G.P. is:

Step 1: Evaluate the inner summation. Using the identity: So the given expression becomes:
Step 2: Simplify the series. This is a geometric series with:
Step 3: Use the sum formula of an infinite GP.
Hence, the required value is

Step 1: Start with the given inequality. Step 2: Transpose the constant term. Step 3: Divide both sides by . (Note: Dividing by a negative number reverses the inequality sign.) Step 4: Write the solution in interval notation. Step 5: Final conclusion. The correct option is .