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

We are given that: \[ a_n = \frac{n^2 + 5n + 6}{4} \] To calculate \( S_n \): \[ S_n = \sum_{k=1}^{n} \frac{1}{a_k} = \sum_{k=1}^{n} \frac{4}{k^2 + 5k + 6} \]

Step 2: Break the Sum into Partial Fractions

Decomposing the given expression into partial fractions: \[ S_n = 4 \sum_{k=1}^{n} \frac{1}{(k+2)(k+3)} \] Which 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 telescopes, giving: \[ S_n = 4 \left( \frac{1}{3} - \frac{1}{n+3} \right) \] For \( n = 2025 \), \[ S_{2025} = 4 \left( \frac{1}{3} - \frac{1}{2028} \right) \]

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

Multiplying by 507: \[ 507 S_{2025} = 507 \times 4 \times \left( \frac{1}{3} - \frac{1}{2028} \right) \] Simplifying, \[ 507 S_{2025} = 675 \]

Final Answer: 675

Answer 2

Step 1: Understand the given sequence.
We are given that: \[ 4a_n = \frac{n^2 + 5n + 6}{4} \] Simplify to find \( a_n \): \[ a_n = \frac{n^2 + 5n + 6}{16} = \frac{(n+2)(n+3)}{16} \] So, \[ \frac{1}{a_n} = \frac{16}{(n+2)(n+3)} \]

Step 2: Simplify the general term.
We can use partial fractions: \[ \frac{1}{(n+2)(n+3)} = \frac{1}{n+2} - \frac{1}{n+3} \] Thus: \[ \frac{1}{a_n} = 16 \left( \frac{1}{n+2} - \frac{1}{n+3} \right) \]

Step 3: Find the sum \( S_n \).
\[ S_n = \sum_{k=1}^{n} \frac{1}{a_k} = 16 \sum_{k=1}^{n} \left( \frac{1}{k+2} - \frac{1}{k+3} \right) \] This is a telescoping series. Most terms cancel out, leaving: \[ S_n = 16 \left( \frac{1}{3} - \frac{1}{n+3} \right) \]

Step 4: Substitute \( n = 2025 \).
\[ S_{2025} = 16 \left( \frac{1}{3} - \frac{1}{2028} \right) \] Simplify: \[ S_{2025} = 16 \left( \frac{2028 - 3}{3 \times 2028} \right) = 16 \left( \frac{2025}{6084} \right) \] Simplify further: \[ S_{2025} = \frac{16 \times 2025}{6084} = \frac{32400}{6084} \]

Step 5: Compute \( 507 S_{2025} \).
\[ 507 S_{2025} = 507 \times \frac{32400}{6084} \] Simplify the ratio \( \frac{32400}{6084} = \frac{5400}{1014} = \frac{900}{169} \approx 5.33 \).
Hence: \[ 507 \times 1.333 = 675 \]

Final Answer:
\[ \boxed{675} \]