If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Solution

Maths sir · 4 steps

Write both four-term sums in terms of d, match them using the one-fifth condition, and d drops out.

1. First four terms. With a = 3, S₄ = 4/2(2a + 3d) = 2(6 + 3d) = 12 + 6d.
2. Next four terms. Each of terms 5 to 8 is exactly 4d more than the matching term among 1 to 4. So their sum is S₄ + 4 × 4d = 12 + 22d.

   12 + 6d = 1/5(12 + 22d) ⇒ 60 + 30d = 12 + 22d ⇒ d = -6

3. Sum of 20 terms. S₂₀ = 20/2(2 × 3 + 19 × (-6)) = 10(6 - 114) = 10 × (-108).
4. So. S₂₀ = -1080, which is option (c).

The idea to keep. The next block of n terms in an A.P. sums to the first block plus n² d, because each term is shifted by nd. That turns block-sum conditions into one quick equation in d.

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