Multiple choice

The sum of the first $50$ terms common to the Arithmetic Sequence $15, 19, 23.....$ And the Arithmetic Sequence $14, 19, 24 ...$ is

  1. $25450$
  2. $24550$
  3. $50900$
  4. $49100$
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A Correct answer
Explanation

The common terms form an arithmetic sequence starting at 19 with a common difference of 20 (LCM of 4 and 5). The 50th term is 19 + (49 * 20) = 19 + 980 = 999. The sum is (n/2) * (first + last) = (50/2) * (19 + 999) = 25 * 1018 = 25450.

AI explanation

The first common term of the two arithmetic sequences is 19, and the common difference of the common terms sequence is the least common multiple of 4 and 5, which is 20. The sequence of common terms becomes 19, 39, 59, and so on. The sum of the first 50 terms of this sequence is found using the formula Sn = n/2 * (2a + (n - 1)d), giving 50/2 * (2*19 + 49*20). Calculating the values inside the parentheses gives 38 + 980 = 1018. Multiplying by 25 yields the total sum of 25450.

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