Answer
$S_{10} = 125$
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 $a_{1}$:
$a_{1} = 3(1) - 4$
Multiply first:
$a_{1} = 3 - 4$
Subtract:
$a_{1} = -1$
Use the explicit formula and plug in $10$ for $n$ to find $a_{10}$:
$a_{10} = 3(10) - 4$
Multiply first:
$a_{10} = 30 - 4$
Subtract:
$a_{10} = 26$
Plug the two values just found into the formula for the sum of a finite arithmetic series, which is given by $S_{n} = \frac{n}{2}(a_{1} + a_{n})$, to find the sum of the series:
$S_{10} = \frac{10}{2}(-1 + 26)$
Evaluate what is inside the parentheses first:
$S_{10} = \frac{10}{2}(25)$
Multiply:
$S_{10} = \frac{250}{2}$
Simplify the fraction:
$S_{10} = 125$
