If aₙ = (- 2)/(4n² - 16n + 15), then a₁ + a₂ + .... + a₂₅ is equal to :

Solution

Maths sir · 4 steps

Factor the denominator into two consecutive odd numbers and the sum collapses to just two terms.

1. Factor. The denominator splits as 4n² - 16n + 15 = (2n-5)(2n-3). These two factors differ by 2.
2. Split the fraction. Because the factors differ by 2, 2/(2n-5)(2n-3) = 1/(2n-5) - 1/(2n-3). With the minus sign in front, this gives a difference of consecutive terms.

   aₙ = 1/(2n-3) - 1/(2n-5)

3. Telescope. Write f(n) = 1/(2n-3). Then f(n-1) = 1/(2n-5), so aₙ = f(n) - f(n-1). Adding from n=1 to 25, every middle term cancels and only f(25) - f(0) is left.
4. So. The sum is 1/47 - 1/-3 = 1/47 + 1/3 = (3 + 47)/141 = 50/141, which is option (c).

The idea to keep. When a quadratic denominator factors into two linear terms that differ by a constant, split it into partial fractions. The sum then telescopes to first-minus-last.

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