In the arithmetic progression (5,16,27,\ldots), the sum of how many first terms will be (1230)?

Correct answer: (15)

The correct answer is option A: 15 terms. In the AP, the first term is \(a=5\) and the common difference is \(d=16-5=11\). The sum formula is \(S_n=\frac{n}{2}[2a+(n-1)d]\). Substitute the values: \(1230=\frac{n}{2}[10+11(n-1)]\). Simplify the bracket: \(10+11n-11=11n-1\), so \(1230=\frac{n(11n-1)}{2}\). Testing the given options is quick. For \(n=15\), the last term is \(5+14(11)=159\), and the sum is \(\frac{15}{2}(5+159)=\frac{15}{2}\times164=1230\). Thus A works. For 16 terms the sum is 1392, for 17 it is 1570.5, and for 18 it is 1764, so B, C and D are wrong. A common mistake is using 16 as the difference; the difference is 11.