Answer
$\sum\limits_{i=1}^{10} -10n + 110$
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:
$d = 90 - 100 = -10$
The common difference $d$ is $-10$.
To find the explicit formula for the $nth$ term, plug in the values we know into the explicit formula for an arithmetic sequence:
$a_{n} = 100 + (n - 1)(-10)$
Use distributive property:
$a_{n} = 100 - 10n + 10$
Combine like factors on the right side of the equation:
$a_{n} = -10n + 110$
Now that we have the explicit formula to find the value of $n$, find the value of $n$ for the last term $10$ by plugging in $10$ into the explicit formula:
$10 = -10n + 110$
Subtract $90$ from each side of the equation:
$-100 = -10n$
Divide each side of the equation by $-10$:
$n = 10$
This means that $10$ is the $10th$ 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:
$\sum\limits_{i=1}^{10} -10n + 110$
