Question

In the following exercises, find each sum.

$\sum _{i=1}^{40}\left(8i-7\right)$

Short answer

The obtained sum is 6280.

Step-by-step solution

Step 1. Given Information.

Consider the given sum: $\sum _{i=1}^{40}\left(8i-7\right)$

simplify the sum.

Express the sum and identify the values.

$\begin{array}{rcl}\sum _{i=1}^{40}\left(8i-7\right)& =& \left(8\left(1\right)-7\right)+\left(8\left(2\right)-7\right)+\left(8\left(3\right)-7\right)+\cdots +\left(8\left(40\right)-7\right)\\ & =& 1+9+17+\cdots +313\\ & =& a_1+a_2+a_3+\cdots +a_n\end{array}$

Step 3. Find the sum.

Substitute the values in some formula $S_n=\frac{n}{2}\left(a_1+a_n\right)$.

$\begin{array}{rcl}{S}_{40}& =& \frac{40}{2}\left(1+313\right)\\ & =& 20\left(314\right)\\ & =& 6280\end{array}$