Question

In the following exercises, find each sum.

$\sum _{i=1}^{25}\left(4i+3\right)$

Short answer

The obtained arithmetic sum is 1375.

Step-by-step solution

Step 1. Given Information.

Consider the given sum:

$\sum _{i=1}^{25}\left(4i+3\right)$

simplify the sum.

Express the sum and identify the values.

$\begin{array}{rcl}\sum _{i=1}^{25}\left(4i+3\right)& =& \left(4\left(1\right)+3\right)+\left(4\left(2\right)+3\right)+\left(4\left(3\right)+3\right)+\cdots +\left(4\left(25\right)+3\right)\\ & =& 7+11+15+\cdots +103\\ & =& a_1+a_2+a_3+\cdots +a_n\end{array}$

Step 3. Find the sum.

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

$\begin{array}{rcl}{S}_{25}& =& \frac{25}{2}\left(7+103\right)\\ & =& \frac{25}{2}\left(110\right)\\ & =& 25\left(55\right)\\ & =& 1375\end{array}$