Question

For positive integers \( n \), if \( 4 a_n = \frac{n^2 + 5n + 6}{4} \) and $$ S_n = \sum_{k=1}^{n} \left( \frac{1}{a_k} \right), \text{ then the value of } 507 S_{2025} \text{ is:} $$ 

• A. 540
• B. 1350
• C. 675
• D. 135

Hint:

When dealing with series sums, consider breaking the series into partial fractions to simplify the terms and cancel out intermediate terms.

Answer 1

The Correct Option is C

Given that: \[ a_n = \frac{n^2 + 5n + 6}{4} \] The sum \( S_n \) is calculated as: \[ S_n = \sum_{k=1}^{n} \frac{1}{a_k} = \sum_{k=1}^{n} \frac{4}{k^2 + 5k + 6} \] Using partial fraction decomposition: \[ S_n = 4 \sum_{k=1}^{n} \frac{1}{(k+2)(k+3)} \] This simplifies to: \[ S_n = 4 \sum_{k=1}^{n} \left( \frac{1}{k+2} - \frac{1}{k+3} \right) \] Thus, the sum becomes a telescoping series: \[ S_n = 4 \left( \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \ldots \right) \] The sum for \( S_n \) cancels out, resulting in: \[ S_{2025} = 4 \left( \frac{1}{3} - \frac{1}{2028} \right) \] Calculating the value of \( 507 S_{2025} \): \[ 507 S_{2025} = 507 \times 4 \times \left( \frac{1}{3} - \frac{1}{2028} \right) = 675 \]