Question

In your own words, explain how to find the sum of the first n terms of an arithmetic sequence without adding all the terms.

Short answer

By using the arithmetic sequence sum formula, we can find the nth term sum.

Step-by-step solution

Step 1. Introduction.

If we have a sequence of nth terms. By substituting the first term and nth term value in the sum formula to find the nth term sum.

Step 2. Explanation.

Consider the sum formula of the nth term.

$S_n=\frac{n}{2}\left(a_1+a_n\right)$

Now, substitute the value of a_n as $a_n=a_1+\left(n-1\right)d$.

$\begin{array}{rcl}{S}_n& =& \frac{n}{2}\left(a_1+a_n\right)\\ & =& \frac{n}{2}\left(a_1+a_1+\left(n-1\right)d\right)\\ & =& \frac{n}{2}\left(2a_1+\left(n-1\right)d\right)\end{array}$

Let's supposed that the first term is 4 and the value of d is 2.

$\begin{array}{rcl}{S}_n& =& \frac{n}{2}\left(2\left(4\right)+\left(n-1\right)2\right)\\ & =& n\left(2+n-1\right)\\ & =& n^2+n\end{array}$