Question

In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.

16, 10, 4, −2, −8, …

Short answer

The obtained sum is -2130.

Step-by-step solution

Step 1. Given Information.

The arithmetic sequence is$16,10,4,-2,-8,\dots$.

Step 2. Find the common difference d.

Find the common difference of consecutive terms.

$\begin{array}{rcl}d& =& 10-16\\ & =& -6\end{array}$

And, now find the 30th term of the sequence.

$\begin{array}{rcl}{a}_{30}& =& 16+\left(30-1\right)\left(-6\right)\\ & =& 16-174\\ & =& -158\end{array}$

Step 3. Use the sum formula.

Use the sum formula $S_n=\frac{n}{2}\left(a_1+a_n\right)$to find the sum of the thirtieth term.

$\begin{array}{rcl}{S}_{30}& =& \frac{30}{2}\left(16-158\right)\\ & =& 15\left(-142\right)\\ & =& -2130\end{array}$