Answer
$\sum\limits_{i=1}^{8} 2n + 5$
Work Step by Step
We want to find the explicit formula for the $nth$ term, so use the explicit formula for an arithmetic sequence, which is given by:
$a_{n} = a_{1} + (n - 1)d$, where $a_{n}$ is a certain term in a sequence, $ a_{1}$ is the first term in the sequence, $n$ is the number of the term within the sequence, and $d$ is the common difference.
The common difference $d$ is found by subtracting one term in the sequence from the one directly following it:
$d = 9 - 7 = 2$
The common difference $d$ is $2$.
To find the explicit formula for the $nth$ term, plug in the values we know into the explicit formula for an arithmetic sequence:
$a_{n} = 7 + (n - 1)(2)$
Use distributive property:
$a_{n} = 7 + 2n - 2$
Combine like factors on the right side of the equation:
$a_{n} = 5 + 2n$
Now that we have the explicit formula to find the value of $n$, find the value of $n$ for the last term $21$ by plugging in $21$ into the explicit formula:
$21 = 2n + 5$
Subtract $5$ from each side of the equation:
$16 = 2n$
Divide each side of the equation by $2$:
$n = 8$
This means that $21$ is the $8th$ and final term in this series.
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:
$\sum\limits_{i=1}^{8} 2n + 5$
