Question

In the following exercises, find each sum.

$\sum _{i=1}^{45}\left(7i-5\right)$

Short answer

The obtained sum is 7020.

Step-by-step solution

Step 1. Given Information.

Consider the given sum:$\sum _{i=1}^{45}\left(7i-5\right)$

simplify the sum.

Express the sum and identify the values.

$\begin{array}{rcl}\sum _{i=1}^{45}\left(7i-5\right)& =& \left(7\left(1\right)-5\right)+\left(7\left(2\right)-5\right)+\left(7\left(2\right)-5\right)+\cdots +\left(7\left(45\right)-5\right)\\ & =& 2+9+16+\cdots +310\\ & =& a_1+a_2+a_3+\cdots +a_n\end{array}$

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{45}& =& \frac{45}{2}\left(2+310\right)\\ & =& \frac{45}{2}\left(312\right)\\ & =& 45\left(156\right)\\ & =& 7020\end{array}$