If the first term of an arithmetic progression is (18), the last term is (126), and there are (19) terms, what is the sum?

Correct answer: (1368)

For an arithmetic progression, the terms increase or decrease by a constant difference. When the first term, last term, and number of terms are known, the sum can be found by pairing terms: the first and last have the same total as the second and second-last, and so on. This gives the efficient formula \(S_n=\frac{n}{2}(a+l)\), where \(a\) is the first term and \(l\) is the last term.

Here, \(n=19\), \(a=18\), and \(l=126\). Substitution gives \(S_{19}=\frac{19}{2}(18+126)=\frac{19}{2}\times144=19\times72=1368\). Since there are an odd number of terms, the pairing idea still works with the middle term left unpaired. Thus the required sum is 1368, so option B is correct. The other values result from an arithmetic or substitution error.