If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :

Solution

Maths sir · 3 steps

The next 15 terms are terms 26 to 40, so add them as their own little A.P.

1. First 25 terms. With a = 3, S₂₅ = 25/2(2 × 3 + 24d) = 25/2(6 + 24d) = 75 + 300d.
2. Next 15 terms. These run from a₂₆ = 3 + 25d to a₄₀ = 3 + 39d. Their sum is 15/2(a₂₆ + a₄₀) = 15/2(6 + 64d) = 45 + 480d.

   75 + 300d = 45 + 480d ⇒ 180d = 30

3. So. d = 30/180 = 1/6, which is option (d).

The idea to keep. For a block of terms in the middle of an A.P., use (number of terms)/2 times (first + last) of that block. Or write it as S₄₀ - S₂₅. Both give the same equation.

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