Arithmetic Series
An arithmetic series is the sum of the values of the arithmetic sequence.
An arithmetic series is the sum of the values of the arithmetic sequence.
Formula for Sum of an Arithmetic Series: \(S_n=\frac{n}{2}(a_1+a_n)\)
In order to find the sum, we must know the number of terms \(n\), the first term \(a_1\), and the last term \(a_n\).
Example 1: Find the sum of the first \(75\) terms of the arithmetic sequence \(a_n=53n−113\).
We know how many terms we are trying to sum \(75\), but we need to find the first term and the seventy-fifth term. We can use the explicit formula for this.
\(\begin{align}&a_1=53(1)−113=−63=−2\ & \ \ &\text{Substitute 1 in for n to find the first term.}\\\\
&a75=53(75)−113=3753−113=3643\ & \ \ &\text{Substitute 75 in for n to find the 75th term.}\end{align}\)
Now using the formula we can find the sum:
\(S_n=\frac{75}{2}(−2+3643)\)
\(S_{75}=4475\)
The sum of the first \(75\) terms is \(4475\).
Example 2: Write the following series using summation notation: \(−10−1+8+17+...+215\).
Example 3: Your starting salary after you graduate college is \($47,000\). You are promised a yearly raise of \($3,000\). If you work at the same job for \(25\) years and receive your promised raise each year, how much total money will you have made during those \(25\) years?
This is an arithmetic series because each term in our sum is \($3,000\) higher than the previous term so \(d=3000\). We also know the first term which is the starting salary so \(a_1=47,000\). We need to find the \(25th\) term in order to find the sum. We will find the explicit formula to find the \(25th\) term.
\(\begin{align}&a_n=47000+3000(n−1)\ & \ \ &\text{1) Substitute}\ a_1\ \text{and d into the explicit formula.}\\\\
&a_n=47000+3000n−3000 \ & \ \ &\text{2) Simplify.}\\\\
&a_n=3000n+44000\ & \ \ &\text{This is the explicit formula.}\\\\
&a_{25}=3000(25)+440000\ & \ \ &\text{3) Substitute 25 into the explicit formula for n.}\\\\
&a_{25}=119000\ & \ \ &\text{4) Solve.}\end{align}\)
The \(25th\) term is \(119,000\) which means you will be making \($119,000\) in the \(25th\) year. We now have enough information to find the sum of those \(25\) years.
\(S_{25}=\frac{25}{2}(47000+119000)\)
\(S_{25}=2075000\)
This means that you will have made a total of \($2,075,000\) during those \(25\) years.
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