In the AP (6,10,14,\ldots), what is the sum from the (4)th term to the (25)th term?

Correct answer: (1320)

The required sum begins with the fourth term and ends with the twenty-fifth term. In an arithmetic progression, the difference between consecutive terms is constant. Here the first term is 6 and the common difference is 4. The first three terms must not be included because the question starts at the fourth term, not at the first term.

The fourth term is 18 and the twenty-fifth term is 102. The number of terms from the fourth through the twenty-fifth is exactly 25 − 4 + 1 = 22. Therefore, their sum is the number of terms multiplied by the average of the first and last terms: \\(S=\frac{22}{2}(18+102)=11\times120=1320\\). Hence option B is correct. Option A would result from an incorrect count or average.