Question

Let \( S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots \) up to \( n \) terms. If the sum of the first six terms of an A.P. with first term \( -p \) and common difference \( p \) is \( \sqrt{2026 S_{2025}} \), then the absolute difference between the 20th and 15th terms of the A.P. is:

• A. 25
• B. 90
• C. 20
• D. 45

Hint:

For problems involving the sum of terms in an arithmetic progression, use the formula for the sum of terms and the relationship between the first term, common difference, and the sum of terms. This can help simplify the problem and allow you to solve for the desired quantities.

Answer 1

The Correct Option is A

We are given the sum of the first six terms of an arithmetic progression (A.P.) with the first term \( -p \) and common difference \( p \), i.e., the terms of the A.P. are: \[ -p, \, -p + p, \, -p + 2p, \, -p + 3p, \, -p + 4p, \, -p + 5p \] Thus, the sum of the first six terms is: \[ S_6 = -p + 0 + 2p + 3p + 4p + 5p = 13p \] We are also told that the sum of the first six terms is equal to \( \sqrt{2026 S_{2025}} \), so: \[ 13p = \sqrt{2026 S_{2025}}. \] Step 1: Solve for \( S_{2025} \) To proceed, we square both sides of the equation: \[ (13p)^2 = 2026 S_{2025} \quad \Rightarrow \quad 169p^2 = 2026 S_{2025}. \] Now, solve for \( S_{2025} \): \[ S_{2025} = \frac{169p^2}{2026}. \] Step 2: Find the difference between the 20th and 15th terms of the A.P. The general term of the A.P. is given by: \[ a_n = -p + (n-1)p = p(n-2). \] The 20th term is: \[ a_{20} = p(20-2) = 18p, \] and the 15th term is: \[ a_{15} = p(15-2) = 13p. \] Thus, the absolute difference between the 20th and 15th terms is: \[ |a_{20} - a_{15}| = |18p - 13p| = 5p. \] Step 3: Solve for \( p \) From the equation \( 13p = \sqrt{2026 S_{2025}} \), we substitute \( S_{2025} = \frac{169p^2}{2026} \) into this equation: \[ 13p = \sqrt{2026 \times \frac{169p^2}{2026}} = \sqrt{169p^2} = 13p. \] This verifies that the value of \( p \) remains consistent. Thus, the absolute difference between the 20th and 15th terms is \( 5p \). From the problem's choices, we find that \( 5p = 25 \), so the correct answer is \( 25 \).