The number of common terms in the progressions
4,9,14,19, … …, up to 25ᵗʰ term and
3,6,9,12, … …, up to 37ᵗʰ term is :

Solution

Maths sir · 3 steps

Common terms of two APs form an AP stepping by the LCM. Stop at the smaller last term.

1. Last terms. First AP: a₂₅ = 4 + 24 × 5 = 124. Second AP: b₃₇ = 3 + 36 × 3 = 111. A common term must lie in both lists, so it cannot exceed 111.
2. AP of common terms. The first number in both lists is 9. After that, common terms repeat every LCM(5, 3) = 15, so they are 9, 24, 39, …

   9 + 15(k-1) ≤ 111 ⇒ k - 1 ≤ 6.8 ⇒ k ≤ 7.8

3. So. k = 7. The common terms are 9, 24, 39, 54, 69, 84, 99, and the next one, 114, is beyond 111. There are 7 common terms, option (d).

The idea to keep. For common terms, find the first shared term, step by the LCM of the two common differences, and stop at the smaller of the two last terms.

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