If the sum of the first 7 terms of an arithmetic progression (AP) is 119 and the sum of the first 17 terms is 714, find the sum of the first \( n \) terms of the AP.

The sum of the first n terms of an arithmetic progression is found using the formula Sn = n/2 (2a + (n-1)d), where a is the first term and d is the common difference. Given the sums of the first 7 and 17 terms, one can solve for a and d, then apply the formula to find Sn for any n.

Explanation:

To find the sum of the first n terms of an arithmetic progression (AP), we need to know the first term (a) and the common difference (d). The formula for the sum of the first n terms is given by Sn = n/2 (2a + (n-1)d).

From the question, we know that the sum of the first 7 terms (S7) is 119, and the sum of the first 17 terms (S17) is 714. Plugging these values into the sum formula, we have two equations:

119 = 7/2 (2a + 6d)
714 = 17/2 (2a + 16d)

Solving this system of equations, we can find the values of a and d. Once a and d are known, we can then use the sum formula to find Sn, the sum of the first n terms for any given n.