What is the formula for the sum of the first n terms in an arithmetic sequence?

The formula for the sum of the first n terms in an arithmetic sequence is derived from the properties of this type of sequence, which has a constant difference between consecutive terms. In an arithmetic sequence, each term can be represented as a = the first term and d = the common difference.

The correct formula is given as (n/2)(2a + (n - 1)d). This formula calculates the sum by leveraging the average of the first and last terms of the sequence. Here’s how it works:

1. The first term is represented as a.

2. The last term in the first n terms can be calculated as a + (n - 1)d.

3. The average of the first and last terms is therefore (a + (a + (n - 1)d))/2 = (2a + (n - 1)d)/2.

4. Since there are n terms, to find the sum of these terms, we multiply the average by n, leading to n * (2a + (n - 1)d)/2, which simplifies to (n/2)(2a + (n - 1)d).

This derivation captures the essence of how the sum of an arithmetic sequence is