Question

Find the sum:$\sum _{i=1}^{35}\left(5i-3\right)$

Short answer

The obtained sum is 3045.

Step-by-step solution

Step 1. Given information.

The given sum expression is$\sum _{i=1}^{35}\left(5i-3\right)$.

simplify the sum. 

Express the sum and identify the values.

$\begin{array}{rcl}\sum _{i=1}^{35}\left(5i-3\right)& =& \left(5\left(1\right)-3\right)+\left(5\left(2\right)-3\right)+\left(5\left(3\right)-3\right)+\cdots +\left(5\left(35\right)-3\right)\\ & =& 2+7+13+\cdots +172\\ & =& a_1+a_2+a_3+\cdots +a_{35}\end{array}$

Thus, the obtained values are $a_1=2$and $a_{35}=172$.

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{35}& =& \frac{35}{2}\left(2+172\right)\\ & =& \frac{35}{2}\left(174\right)\\ & =& 35\left(87\right)\\ & =& 3045\end{array}$