Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ________.

Solution

Maths sir · 4 steps

The common terms form their own AP: find its first term and difference, then stop at the smaller last term.

1. Last terms. The first series ends at 3 + 77 × 3 = 234. The second ends at 5 + 58 × 4 = 237. A common term must lie in both, so it can be at most 234.
2. Common AP. The first common term is 9. The common terms repeat every LCM(3, 4) = 12, so they are 9, 21, 33, …
3. Count them. Solve 9 + (k-1) × 12 ≤ 234, which gives k - 1 ≤ 18.75. So k = 19 terms, and the last one is 9 + 18 × 12 = 225.

   S = 19/2(9 + 225) = 19 × 117 = 2223

4. So. The sum of the common terms is 2223.

The common slip: 2460. Using 237 as the limit wrongly counts 237 as common, but 237 is beyond the first series. That gives 20 terms and a sum of 2460.

The idea to keep. Terms common to two APs form a new AP with difference equal to the LCM of the two differences. Its limit is the smaller of the two last terms.

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