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# Sum of the First $n$ Terms of an Arithmetic Sequence

Suppose a sequence of numbers is arithmetic (that is, it increases or decreases by a constant amount each term), and you want to find the sum of the first $n$ terms.

Denote this partial sum by ${S}_{n}$ . Then

${S}_{n}=\frac{n\left({a}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{a}_{n}\right)}{2}$ ,
where $n$ is the number of terms, ${a}_{1}$ is the first term and ${a}_{n}$ is the last term.

The sum of the first $n$ terms of an arithmetic sequence is called an arithmetic series .

Example 1:

Find the sum of the first $20$ terms of the arithmetic series if ${a}_{1}=5$ and ${a}_{20}=62$ .

$\begin{array}{l}{S}_{20}=\frac{20\left(5\text{\hspace{0.17em}}+\text{\hspace{0.17em}}62\right)}{2}\\ {S}_{20}=670\end{array}$

Example 2:

Find the sum of the first $40$ terms of the arithmetic sequence $2,5,8,11,\cdots$ .

First find the ${40}^{\text{th}}$ term:

$\begin{array}{l}{a}_{40}={a}_{1}+\left(n-1\right)d\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=2+39\left(3\right)=119\end{array}$

Then find the sum:

$\begin{array}{l}{S}_{n}=\frac{n\left({a}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{a}_{n}\right)}{2}\\ {S}_{40}=\frac{40\left(2\text{\hspace{0.17em}}+\text{\hspace{0.17em}}119\right)}{2}=2420\end{array}$

Example 3:

Find the sum:

$\underset{k=1}{\overset{50}{\sum }}\left(3k+2\right)$

First find ${a}_{1}$ and ${a}_{50}$ :

$\begin{array}{l}{a}_{1}=3\left(1\right)+2=5\\ {a}_{50}=3\left(50\right)+2=152\end{array}$

Then find the sum:

$\begin{array}{l}{S}_{k}=\frac{k\left({a}_{1}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}{a}_{k}\right)}{2}\\ {S}_{50}=\frac{50\left(5\text{\hspace{0.17em}}+\text{\hspace{0.17em}}152\right)}{2}=3925\end{array}$

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