Multiple choice

The sum of $n$ terms of an arithmetic series is $S_n = 2n - n^2$. Find the first term and the common difference.

  1. $a = 2; d = 2$
  2. $a = 0; d = 1$
  3. $a = 1; d = -2$
  4. $a = -1; d = 2$
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C Correct answer
Explanation

S1 = a = 2(1) - 1^2 = 1. S2 = a1 + a2 = 2(2) - 2^2 = 0. So a2 = -1. Common difference d = a2 - a1 = -1 - 1 = -2.

AI explanation

The first term of the series is found by substituting n = 1 into the sum formula, giving a = 2(1) - (1)^2, which equals 1. The sum of the first two terms is found by substituting n = 2, giving 2(2) - (2)^2, which equals 0. Since the first term is 1, the second term must be -1, making the common difference -1 - 1, which is -2. Therefore, the first term is 1 and the common difference is -2.

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