Question

In the following exercises, find each sum.

$\sum _{i=1}^{30}\left(-5i+1\right)$

Short answer

The obtained arithmetic sum is -2325.

Step-by-step solution

Step 1. Given Information.

Consider the given sum:

$\sum _{i=1}^{30}\left(-5i+1\right)$

simplify the sum.

Express the sum and find identify values.

$\begin{array}{rcl}\sum _{i=1}^{30}\left(-5i+1\right)& =& \left(-5\left(1\right)+1\right)+\left(-5\left(2\right)+1\right)+\left(-5\left(3\right)+1\right)+\cdots +\left(-5\left(30\right)+1\right)\\ & =& \left(-4\right)+\left(-9\right)+\left(-14\right)+\cdots +\left(-149\right)\\ & =& a_1+a_2+a_3+\cdots +a_n\end{array}$

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{30}& =& \frac{30}{2}\left(-4-149\right)\\ & =& 15\left(-155\right)\\ & =& -2325\end{array}$