Question

Find the sum:$\sum _{i=1}^{30}\left(6i-4\right)$

Short answer

The obtained sum is 2670.

Step-by-step solution

Step 1. Given Information.

Consider the given sum:$\sum _{i=1}^{30}\left(6i-4\right)$

simplify the sum.

Express the sum and identify the values.

$\begin{array}{rcl}\sum _{i=1}^{30}\left(6i-4\right)& =& \left(6\left(1\right)-4\right)+\left(6\left(2\right)-4\right)+\left(6\left(3\right)-4\right)+\cdots +\left(6\left(30\right)-4\right)\\ & =& 2+8+14+\cdots +176\\ & =& a_1+a_2+a_3+\cdots +a_{30}\end{array}$

Thus, the obtained values are$a_1=2$and$a_{30}=176$.

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{30}& =& \frac{30}{2}\left(2+176\right)\\ & =& 15\left(178\right)\\ & =& 2670\end{array}$