Question

In the following exercises, find each sum.

$\sum _{i=1}^{35}\left(-6i-2\right)$

Short answer

The obtained arithmetic sum is -3850.

Step-by-step solution

Step 1. Given Information.

Consider the given sum:

$\sum _{i=1}^{35}\left(-6i-2\right)$

simplify the sum. 

Express the sum and find identify values.

$\begin{array}{rcl}\sum _{i=1}^{35}\left(-6i-2\right)& =& \left(-6\left(1\right)-2\right)+\left(-6\left(2\right)-2\right)+\left(-6\left(3\right)-2\right)+\cdots +\left(-6\left(35\right)-2\right)\\ & =& -8-14-20-\cdots -212\\ & =& a_1+a_2+a_3+\cdots +a_n\end{array}$

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{35}& =& \frac{35}{2}\left(-8-212\right)\\ & =& \frac{35}{2}\left(-220\right)\\ & =& -35\left(110\right)\\ & =& -3850\end{array}$