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Just as we studied special types of sequences, we will look at special types of series. Recall that an arithmetic sequence is a sequence in which the difference between any two consecutive terms is the common difference , $d.$ The sum of the terms of an arithmetic sequence is called an arithmetic series . We can write the sum of the first $n$ terms of an arithmetic series as:
We can also reverse the order of the terms and write the sum as
If we add these two expressions for the sum of the first $n$ terms of an arithmetic series, we can derive a formula for the sum of the first $n$ terms of any arithmetic series.
Because there are $n$ terms in the series, we can simplify this sum to
We divide by 2 to find the formula for the sum of the first $n$ terms of an arithmetic series.
An arithmetic series is the sum of the terms of an arithmetic sequence. The formula for the sum of the first $n$ terms of an arithmetic sequence is
Given terms of an arithmetic series, find the sum of the first $n$ terms.
Find the sum of each arithmetic series.
We are given ${a}_{1}=5$ and $\text{\hspace{0.17em}}{a}_{n}=32.$
Count the number of terms in the sequence to find $n=10.$
Substitute values for $\text{\hspace{0.17em}}{a}_{1},{a}_{n}\text{\hspace{0.17em},}$ and $n$ into the formula and simplify.
We are given ${a}_{1}=20$ and ${a}_{n}=-50.$
Use the formula for the general term of an arithmetic sequence to find $n.$
Substitute values for
${a}_{1},{a}_{n}\text{,}\text{\hspace{0.17em}}n$ into the formula and simplify.
To find ${a}_{1},\text{\hspace{0.17em}}$ substitute $k=1$ into the given explicit formula.
We are given that $n=12.$ To find ${a}_{12},\text{\hspace{0.17em}}$ substitute $k=12$ into the given explicit formula.
Substitute values for ${a}_{1},{a}_{n},$ and $n$ into the formula and simplify.
Use the formula to find the sum of each arithmetic series.
$\text{1}\text{.4+1}\text{.6+1}\text{.8+2}\text{.0+2}\text{.2+2}\text{.4+2}\text{.6+2}\text{.8+3}\text{.0+3}\text{.2+3}\text{.4}$
$\text{26}\text{.4}$
On the Sunday after a minor surgery, a woman is able to walk a half-mile. Each Sunday, she walks an additional quarter-mile. After 8 weeks, what will be the total number of miles she has walked?
This problem can be modeled by an arithmetic series with $\text{\hspace{0.17em}}{a}_{1}=\frac{1}{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}d=\frac{1}{4}.\text{\hspace{0.17em}}$ We are looking for the total number of miles walked after 8 weeks, so we know that $n=8\text{,}$ and we are looking for $\text{\hspace{0.17em}}{S}_{8}.\text{\hspace{0.17em}}$ To find ${a}_{8},$ we can use the explicit formula for an arithmetic sequence.
We can now use the formula for arithmetic series.
She will have walked a total of 11 miles.
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