Answer
$\sum\limits_{i=1}^{8} (-2n + 23)$
$S_{8} = 112$
Work Step by Step
\ltTo 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 following term:
$d = 19 - 21 = -2$
Plug values into the explicit formula for an arithmetic sequence to find the explicit formula of this series:
$a_n = 21 + (n - 1)(-2)$
Use distributive property:
$a_n = 21 - 2n + 2$
Combine like terms:
$a_n = -2n + 23$
Now, we can plug our values into the summation notation, where we plug in the explicit formula for finding the $nth$ term with the lower limit being $1$ and the upper limit being the value of $n$ for the last term in the series:
$\sum\limits_{i=1}^{8} (-2n + 23)$
To find the final term in the series, use the explicit formula for a finite arithmetic sequence:
$a_{8} = 21 + (8 - 1)(-2)$
Evaluate expressions within parentheses first:
$a_{8} = 21 + (7)(-2)$
Multiply:
$a_{8} = 21 - 14$
Add to solve:
$a_{8} = 7$
Now we can plug the values that are known into the formula to find the sum of a finite arithmetic series:
$S_{8} = \frac{8}{2}(21 + 7)$
Evaluate what is in parentheses first:
$S_{8} = \frac{8}{2}(28)$
Multiply:
$S_{8} = \frac{224}{2}$
Simplify the fraction:
$S_{8} = 112$
