Mathematics
Mathematics
Problem
The first term of an arithmetic progression is and the common difference is . Find the sum of its first terms.Intuitive explanation
An arithmetic progression grows by a constant amount each term, so you can write out the sequence quickly—but adding 20 terms individually would be tedious. The key insight is that in any AP, the first and last terms are equidistant from the middle, so they pair up symmetrically: the sum of the first and last term equals the sum of the second and second-to-last term, and so on. Use this pairing to collapse the sum into a simple formula.
Intuitive explanation
An arithmetic progression grows by a constant amount each term, so you can write out the sequence quickly—but adding 20 terms individually would be tedious. The key insight is that in any AP, the first and last terms are equidistant from the middle, so they pair up symmetrically: the sum of the first and last term equals the sum of the second and second-to-last term, and so on. Use this pairing to collapse the sum into a simple formula.