What is the sum of the first (23) terms of the AP (19,30,41,\ldots)?

Correct answer: (3220)

The direct answer is option A: 3220. In the AP 19, 30, 41, …, the first term is a=19 and the common difference is d=30−19=11. The number of terms is n=23. Apply \\(S_n=\\frac{n}{2}[2a+(n-1)d]\\). First, n−1=22. Then (n−1)d=22×11=242. Also, 2a=2×19=38. Add inside the bracket: 38+242=280. Therefore \\(S_{23}=\\frac{23}{2}×280=23×140=3220\\). Hence option A is correct. Option B, 3197, is wrong because it is 23 less than the correct sum and does not result from the formula. Option C, 3243, is wrong because it is 23 more than the correct sum, suggesting an incorrect extra amount. Option D, 3266, is also wrong because it uses an incorrect final total or arithmetic step. A useful check is to find the last term: \\(a_{23}=19+22×11=261\\). The average of the first and last terms is \\((19+261)/2=140\\), so the sum is 23×140=3220. The fact that n is odd causes no difficulty; the standard formula works for every positive whole number of terms. Exam cue: identify a, d and n before substituting, and do not forget that n−1=22, not 23.