Answer
$S_7 = 455$
Work Step by Step
To find the sum of a finite series, we can use the formula:
$S_n = \frac{n}{2}(a_1 + a_n)$, where $S_n$ is the sum of the series, $n$ is the number of terms in the series, $a_1$ is the first term, and $a_n$ is the $nth$ term.
We already have $a_1$ and $n$; however, we need to find $a_n$. This can be done by using the explicit formula for arithmetic sequences, which is given by $a_n = a + (n - 1)d$. We can find $d$ by subtracting one term from the term following it:
$d = 55 - 50 = 5$
Plug values into the explicit formula:
$a_7 = 50 + (7 - 1)(5)$
Evaluate expressions within parentheses first:
$a_7 = 50 + (6)(5)$
Multiply:
$a_7 = 50 + (30)$
Add to solve:
$a_7 = 80$
Now we can plug the values that are known into the formula to find the sum of a finite arithmetic series:
$S_7 = \frac{7}{2}(50 + 80)$
Evaluate what is in parentheses first:
$S_7 = \frac{7}{2}(130)$'
Multiply:
$S_7 = \frac{910}{2}$
Simplify the fraction:
$S_7 = 455$
