1/(3² - 1) + 1/(5² - 1) + 1/(7² - 1) + .... + 1/((201)² - 1) is equal to

Solution

Maths sir · 4 steps

Factor each denominator as a difference of squares, split into partial fractions, and watch it collapse.

1. General term. The odd numbers run 3, 5, 7, …, 201, so write each one as 2r+1 with r = 1 to 100. Then (2r+1)² - 1 = 4r² + 4r = 4r(r+1).
2. Partial fractions. Since 1/r(r+1) = 1/r - 1/(r+1), every term splits into a difference of two neighbouring fractions.

   S = 1/4Σᵣ₌₁¹⁰⁰(1/r-1/(r+1))

3. Telescope. Each -1/(r+1) cancels the +1/(r+1) in the next bracket. Only the first piece 1 and the last piece -1/101 survive.
4. So. S = 1/4(1 - 1/101) = 1/4 × 100/101 = 25/101, which is option (b).

The idea to keep. When a denominator is a product of two terms a fixed gap apart, split it into partial fractions. The middle terms cancel, so you only need the first and last pieces.

Answer from the NTA final answer key. Solution by PrepWiser.