Answer
$S_{5} = 25$
Work Step by Step
In order to find the sum of this finite series, we need to find the first term, $a_{1}$ and the last term, $a_{5}$.
First, use the explicit formula given and plug in $1$ for $n$ to find $a_{1}$:
$a_{1} = 2(1) - 1$
Multiply first:
$a_{1} = 2 - 1$
Subtract:
$a_{1} = 1$
Use the explicit formula and plug in $5$ for $n$ to find $a_{5}$:
$a_{5} = 2(5) - 1$
Multiply first:
$a_{5} = 10 - 1$
Subtract:
$a_{5} = 9$
Plug the two values just found into the formula for the sum of a finite arithmetic series, which is given by $S_{n} = \frac{n}{2}(a_{1} + a_{n})$:
$S_{5} = \frac{5}{2}(1 + 9)$
Evaluate what is inside the parentheses first:
$S_{5} = \frac{5}{2}(10)$
Multiply:
$S_{5} = \frac{50}{2}$
Simplify the fraction:
$S_{5} = 25$
