The sum of the first three terms of an arithmetic sequence is 27. The second term is[Fill in the blank]

Question

BlackTom AI · Accepted Answer

The problem states: The sum of the first three terms of an arithmetic sequence is 27. The second term is ______.
Option A: 4.5 — If a2 were 4.5, then a1 + a2 + a3 would equal 3a2 = 13.5, not 27, so this cannot be correct.
Option B: 6 — If a2 were 6, then the sum a1 + a2 + a3 would be 3a2 = 18, which does not match 27, so this is false.
Option C: 9 — In an arithmetic sequence, a2 = (a1 + a3)/2, and the sum a1 + a2 + a3 = 3a2, since a1 + a3 = 2a2. Given the total is 27, 3a2 = 27, so a2 = 9. This matches the given sum, making this option consistent.
Option D: 13.5 — If a2 were 13.5, the sum a1 + a2 + a3 would be 3a2 = 40.5, which contradicts the stated sum of 27, so this is incorrect.
To summarize the reasoning: S = a1 + a2 + a3 = 3a2 for any arithmetic sequence, because a1, a2, a3 are consecutive terms with equal differences. Since S = 27, a2 = 27/3 = 9. Therefore, the correct choice corresponds to a2 = 9, which is option C.

The sum of the first three terms of an arithmetic sequence is 27. The second term is[Fill in the blank]

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