JEE Main 2022 · 25 July · evening · Single correct · +4 / −1
The sum Σₙ₌₁²¹ 3/(4n - 1)(4n + 3) is equal to
- (a) 7/87
- (b) 7/29
- (c) 14/87
- (d) 21/29
Answer and solution
Answer (b) 7/29
Solution
Maths sir · 3 steps
The two factors differ by 4, so split into partial fractions and let the middle terms cancel.
- Split the term. The factors 4n+3 and 4n-1 differ by 4. So 1/(4n-1)(4n+3) = 1/4(1/(4n-1) - 1/(4n+3)). With the 3 on top, each term is 3/4(1/(4n-1) - 1/(4n+3)).
- Watch it telescope. Writing out the brackets gives (1/3-1/7) + (1/7-1/11) + … + (1/83-1/87). Every middle fraction cancels, leaving only the first and the last, because 4(21)+3 = 87.
Σₙ₌₁²¹ 3/(4n-1)(4n+3) = 3/4(1/3 - 1/87) = 3/4 × 28/87
- So. 3/4 × 28/87 = 84/348 = 7/29, which is option (b).
The idea to keep. For 1/(an+b)(an+b+d), split as 1/d(1/(an+b) - 1/(an+b+d)). Divide by the gap d, then only the first and last pieces survive.
Answer from the NTA final answer key. Solution by PrepWiser.