Question

The sum of the first \( n \) terms of a geometric progression is 255, the \( k \)-th term is 128, and the common ratio is 2. The value of \( k \) satisfies the equation:

• A. \( 2^k - 7k = 8 \)
• B. \( k^2 - 7k = 8 \)
• C. \( 2k^2 - 17k = 7 \)
• D. \( 2k^2 - 15k = 9 \)

Hint:

For geometric progressions, the general term and sum formulas are very useful. Solve for the term values and use the known sum to find the required term index.

Answer 1

The Correct Option is A

We know that the sum of the first \( n \) terms of a geometric progression is given by: \[ S_n = a \frac{1 - r^n}{1 - r} \] where \( r \) is the common ratio and \( a \) is the first term. From the given data, set up the equation for the sum of the first \( n \) terms and the \( k \)-th term, and solve for \( k \).