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

Step 1: Identify the Given Sequence

The given sequence is: \[ a_n = \frac{n^2 + 5n + 6}{4} \] The sum \( S_n \) is defined as: \[ S_n = \sum_{k=1}^{n} \frac{1}{a_k} = \sum_{k=1}^{n} \frac{4}{k^2 + 5k + 6} \]

Step 2: Break the Sum into Partial Fractions

Decompose the expression into partial fractions: \[ S_n = 4 \sum_{k=1}^{n} \frac{1}{(k+2)(k+3)} \] This simplifies to: \[ S_n = 4 \sum_{k=1}^{n} \left( \frac{1}{k+2} - \frac{1}{k+3} \right) \]

Step 3: Evaluate the Series

The series is a telescoping series, yielding: \[ S_n = 4 \left( \frac{1}{3} - \frac{1}{n+3} \right) \] For \( n = 2025 \): \[ S_{2025} = 4 \left( \frac{1}{3} - \frac{1}{2028} \right) \]

Step 4: Compute \( 507 \times S_{2025} \)

Multiply \( S_{2025} \) by 507: \[ 507 S_{2025} = 507 \times 4 \times \left( \frac{1}{3} - \frac{1}{2028} \right) \] The simplified result is: \[ 507 S_{2025} = 675 \]

Final Answer: 675