Question

In the following exercises, find each sum.

$\sum _{i=1}^{50}\left(3i+6\right)$

Short answer

The obtained sum is 4125.

Step-by-step solution

Step 1. Given Information.

Consider the given sum:

$\sum _{i=1}^{50}\left(3i+6\right)$

simplify the sum.

Express the sum and identify the values.

$\begin{array}{rcl}\sum _{i=1}^{50}\left(3i+6\right)& =& \left(3\left(1\right)+6\right)+\left(3\left(2\right)+6\right)+\left(3\left(3\right)+6\right)+\cdots +\left(3\left(50\right)+6\right)\\ & =& 9+12+15+\cdots +156\\ & =& a_1+a_2+a_3+\cdots +a_{50}\end{array}$

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{50}& =& \frac{50}{2}\left(9+156\right)\\ & =& 25\left(165\right)\\ & =& 4125\end{array}$