5.03 Arithmetic sequences

What is an arithmetic sequence?

A sequence in which each term changes from the last by adding or subtracting a constant amount is called an arithmetic sequence. The number being added or subtracted to produce the next number in the sequence is known as the common difference, which will result from subtracting any two successive terms $u_{n+1}-u_n$un+1​−un​.

For example. the sequence $-3,5,13,21,\ldots$−3,5,13,21,… is an arithmetic sequence with a common difference of $8$8. On the other hand, the sequence $1,10,100,1000,\ldots$1,10,100,1000,… is not arithmetic because the difference between each term is not constant.

The first term in an arithmetic sequence is denoted by the letter $a$a and the common difference is denoted by $d$d. Since, $u_1=u_0+d$u1​=u0​+d, $u_2=u_1+d$u2​=u1​+d and so on, any arithmetic sequence can be expressed as the recurrence relation that was defined in the last lesson of this chapter:

$u_{n+1}=u_n+d,u_0=a$un+1​=un​+d,u0​=a

An explicit generating rule can be found in terms of $a$a and $d$d, this is useful for finding the $n$nth term without listing the sequence or having to use the previous term in the sequence each time to find the next term.

Consider the following table to see the pattern for the explicit formula. For the sequence $-3,5,13,21,\ldots$−3,5,13,21,…, the starting term is $-3$−3 and there is a common difference of $8$8, that is $a=-3$a=−3 and $d=8$d=8. A table of the sequence is show below:

$n$n	$u_n$un​	Pattern
$0$0	$-3$−3	$-3$−3
$1$1	$5$5	$-3+8$−3+8
$2$2	$13$13	$-3+2\times8$−3+2×8
$3$3	$21$21	$-3+3\times8$−3+3×8
$\ldots$…
$n$n	$u_n$un​	$-3+n\times8$−3+n×8

By correctly identifying the pattern, the tenth term becomes $u_{10}=77=-3+10\times8$u10​=77=−3+10×8 and the one-hundredth term would be $u_{100}=797=-3+100\times8$u100​=797=−3+100×8. Following the pattern, the explicit formula for the $n$nth term is $u_n=-3+n\times8$un​=−3+n×8.

A similar table can be created for any arithmetic sequence with starting value $a$a and common difference $d$d and the same pattern would be observed. Hence, the explicit generating rule for the $n$nth term in any arithmetic sequence is given by:

$u_n=a+n\times d$un​=a+n×d

Note that sometimes arithmetic sequences are presented with $u_1=a$u1​=a as the first term. In these cases, the explicit form is given by $u_n=a+\left(n-1\right)\times d$un​=a+(n−1)×d instead.

 

Worked examples

Example 1

For the sequence $87,80,73,66...$87,80,73,66..., find and explicit rule for the $n$nth term and hence, find the $30$30th term.

Think: Check that the sequence is arithmetic, does each term differ from the last by a constant? Then write down the the starting value $a$a and common difference $d$d and substitute these into the general form: $u_n=a+nd$un​=a+nd

Do: Each term is a decrease from the last by $7$7. So we have an arithmetic sequence with: $a=87$a=87 and $d=-7$d=−7. The general formula for this sequence is: $u_n=87+n\times\left(-7\right)$un​=87+n×(−7) or $u_n=87-7n$un​=87−7n.

Since the first term is $u_0$u0​, the $30$30th term will be $u_{29}$u29​. Hence, the $30$30th term is: $u_{30}=87-7\times29=-116$u30​=87−7×29=−116.

Example 2

For the sequence $10,14,18,22,26,...$10,14,18,22,26,..., find $n$n if the $n$nth term is $186$186.

Think: Find a general rule for the sequence, substitute in $186$186 for $u_n$un​ and rearrange for $n$n.

Do: This is an arithmetic sequence with $a=10$a=10 and common difference $d=4$d=4. Hence, the general rule is: $u_n=10+n\times4$un​=10+n×4, or just $u_n=6+4n$un​=6+4n. Substituting $u_n=186$un​=186, we get:

$186$186	$=$=	$10+4n$10+4n
$\therefore4n$∴4n	$=$=	$176$176
$n$n	$=$=	$44$44

$n=44$n=44 corresponds to the $45$45th term in the sequence, and so the $45$45th term in the sequence is $186$186.

Example 3

If an arithmetic sequence has $u_5=38$u5​=38 and $u_9=66$u9​=66, find the recurrence relation for the sequence.

• Think: To find the recurrence relation we need the starting value and common difference. As we have two terms we can set up two equations in terms of $u_0$u0​ and $d$d using $u_{n+1}=u_n+d$un+1​=un​+d.

Do:

$u_5$u5​:

$u_0+4d=38$u0​+4d=38

$.....\left(1\right)$.....(1)

and

$u_9$u9​:

$u_0+8d=66$u0​+8d=66

$.....\left(2\right)$.....(2)

 

If we now subtract equation $\left(1\right)$(1) from equation $\left(2\right)$(2) the first term in each equation will cancel out to leave us with:

$\left(8d-4d\right)$(8d−4d)	$=$=	$66-38$66−38
$4d$4d	$=$=	$28$28
$\therefore d$∴d	$=$=	$7$7

With the common difference found to be $7$7, then we know that, using equation $\left(1\right)$(1) $u_0+4\times7=38$u0​+4×7=38 and so $u_0$u0​ is $10$10. The recurrence relation for this sequence is given by:

$u_{n+1}=u_n+7,u_0=10$un+1​=un​+7,u0​=10

Practice questions

QUESTION 1

QUESTION 2

 

Arithmetic sequences in tables and graphs

Any arithmetic sequence in the general form given by $u_n=a+n\times d$un​=a+n×d, is in the form of the equation of a straight line $\left(y=mx+c\right)$(y=mx+c). So if an arithmetic sequence is plotted as a series of points, all the points lie on a straight line with the slope being the common difference. This makes sense as there is a constant rate of change, i.e. the common difference.

 

The term represented by the point shown at $n=1$n=1 is $u_1=5$u1​=5 and the slope of this line is the common difference $d=2$d=2.

An arithmetic sequence can be identified from a table, such as:

$n$n	$1$1	$2$2	$3$3	$4$4	$5$5
$u_n$un​	$5$5	$7$7	$9$9	$11$11	$13$13

Here, the term $u_1=5$u1​=5 and the common difference can be seen by comparing adjacent $u_n$un​ values in the second row. Since $2$2 is being added each time to create the next term in the sequence, the common difference is $d=2$d=2.

 

Practice questions

QUESTION 3