The average of the first (30) terms of an AP is (76). What is the value of (S_{30})?

Correct answer: (2280)

Direct answer: B, 2280. Average means total sum divided by the number of terms: \(\text{average}=\frac{S_n}{n}\). Rearranging gives \(S_n=n\times\text{average}\). The question gives 30 terms and average 76, so \(S_{30}=30\times76=2280\). This rule works for an AP and, more generally, for any finite list of numbers. Option B is therefore correct. Option A, 2160, would correspond to an average of 72 for 30 terms, not 76. Option C, 2340, would require an average of 78. Option D, 2400, would require an average of 80. No common difference is needed because the average is already supplied. A useful AP check is that the average of the first and last terms also equals 76, but the direct multiplication is enough here. Memory cue: total equals number of entries multiplied by average; do not divide again.