Multiple choice

The number of common terms between the sequences given by $1,4,7,10,.....298$ and $2,4,6,8,.....300$ is

  1. $49$
  2. $50$
  3. $48$
  4. $100$
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B Correct answer
Explanation

Sequence 1: 1, 4, 7, ..., 298 (Arithmetic, a=1, d=3). Sequence 2: 2, 4, 6, ..., 300 (Arithmetic, a=2, d=2). Common terms must be even and in sequence 1. Sequence 1 terms: 3n-2. 3n-2 must be even, so 3n must be even, n must be even. n=2, 4, ..., 100. Terms are 4, 10, 16, ..., 298. This is an AP with a=4, d=6. 298 = 4 + (k-1)6 => 294 = (k-1)6 => 49 = k-1 => k=50.

AI explanation

The first sequence is an arithmetic progression with a common difference of 3, so its common terms with even numbers must be multiples of both 2 and 3. The least common multiple of 2 and 3 is 6, meaning the common terms form the sequence 4, 10, 16, and so forth, which is an arithmetic progression with a first term of 4 and a common difference of 6. The general term is 6k minus 2, and solving 6k minus 2 equals 298 gives k equals 50. The result is 50.

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