Sum of arithmetic series 3, 7, 11, ..., with n terms. Provide S_n.

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Multiple Choice

Sum of arithmetic series 3, 7, 11, ..., with n terms. Provide S_n.

Explanation:
For an arithmetic series, the sum of the first n terms is given by S_n = n/2 [2a1 + (n−1)d], where a1 is the first term and d is the common difference. Here, a1 = 3 and d = 4, so a_n = a1 + (n−1)d = 3 + 4(n−1) = 4n − 1. Then S_n = n/2 [a1 + a_n] = n/2 [3 + (4n − 1)] = n/2 (4n + 2) = n(2n + 1). So the sum is S_n = n(2n + 1). Verification with small n (n = 1 → 3; n = 2 → 10) confirms the result.

For an arithmetic series, the sum of the first n terms is given by S_n = n/2 [2a1 + (n−1)d], where a1 is the first term and d is the common difference. Here, a1 = 3 and d = 4, so a_n = a1 + (n−1)d = 3 + 4(n−1) = 4n − 1. Then

S_n = n/2 [a1 + a_n] = n/2 [3 + (4n − 1)] = n/2 (4n + 2) = n(2n + 1).

So the sum is S_n = n(2n + 1). Verification with small n (n = 1 → 3; n = 2 → 10) confirms the result.

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