If the sum of the first 7 terms of an arithmetic progression (AP) is 49, and the sum of the first 17 terms is 289, find the sum of the first n terms of the AP.

The sum of the first n terms of the arithmetic progression (AP) can be found using the formula Sn = n/2 * [2a + (n - 1)d], where a is the first term and d is the common difference.

Explanation:

Given the sum of the first 7 terms of the AP is 49 and the sum of the first 17 terms is 289, we can set up two equations based on the sum formula. For the first equation, substituting n = 7 and Sn = 49, we get 49 = 7/2 * [2a + 6d]. For the second equation, with n = 17 and Sn = 289, we have 289 = 17/2 * [2a + 16d].

Solving these simultaneous equations for a and d, we find a = 1 and d = 6. Now, using the formula for the sum of the first n terms, Sn = n/2 * [2a + (n - 1)d], we can find the sum of the first n terms of the AP for any given value of n.

This method provides a systematic approach to calculating the sum of an arithmetic progression, ensuring accuracy and efficiency in solving such problems. Additionally, understanding the underlying principles of arithmetic progressions enables us to apply these concepts in various mathematical contexts, enhancing problem-solving skills and mathematical proficiency.