Arithmetic Sequences

An arithmetic sequence is a sequence of numbers which increases or decreases by a constant amount each term.

We can write a formula for the $n^{th}$ term of an arithmetic sequence in the form

$a_{n} = d n + c$ ,

where $d$ is the common difference . Once you know the common difference, you can find the value of $c$ by plugging in $1$ for $n$ and the first term in the sequence for $a_{1}$ .

Example 1:

${1, 5, 9, 13, 17, 21, 25, ...}$

is an arithmetic sequence with common difference of $4$ .

(Since

$5 - 1 = 4$ ,

$9 - 5 = 4$ ,

etc.)

To find the next $3$ terms, we just keep adding $4$ :

$\begin{array}{l} 25 + 4 = 29 \\ 29 + 4 = 33 \\ 33 + 4 = 37 \end{array}$

So, the next $3$ terms are $29$ , $33$ , and $37$ .

To find a formula for the $n^{th}$ term, we substitute $n = 1, a_{1} = 1$ and $d = 4$ in

$a_{n} = d n + c$

to find $c$ .

$\begin{array}{l} 1 = 4 (1) + c \\ c = - 3 \end{array}$

So, a formula for the $n^{th}$ term of the sequence is

$a_{n} = 4 n - 3$ .

Example 2:

${12, 9, 6, 3, 0, - 3, - 6, ...}$

is an arithmetic sequence with common difference of $- 3$ .

(Since

$9 - 12 = - 3$

$6 - 9 = - 3$

etc. Note that since the sequence is decreasing, the common difference is negative.)

To find the next $3$ terms, we just keep subtracting $3$ :

$\begin{array}{l} - 6 - 3 = - 9 \\ - 9 - 3 = - 12 \\ - 12 - 3 = - 15 \end{array}$

So, the next $3$ terms are $- 9$ , $- 12$ , and $- 15$ .

To find a formula for the $n^{th}$ term, we substitute $n = 1, a_{1} = 12$ , and $d = - 3$ in

$a_{n} = d n + c$

to find $c$ .

$\begin{array}{l} 12 = - 3 (1) + c \\ c = 15 \end{array}$

So, a formula for the $n^{th}$ term of this sequence is

$a_{n} = - 3 n + 15$ .

Example 3:

${2, 3, 5, 8, 12, 17, 23, ...}$

is not an arithmetic sequence. The difference $a_{2} - a_{1}$ is $1$ , but the next difference $a_{3} - a_{2}$ is $2$ .

No formula of the form

$a_{n} = d n + c$ can be written for this sequence.

Arithmetic Sequences

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