A sequence in which each term changes from the last by adding a constant amount is called an arithmetic sequence. We refer to the constant the terms are changing by as the common difference, which will result from subtracting any two successive terms $\left(t_{n+1}-t_n\right)$(tn+1−tn).
The progression $-3,5,13,21,\ldots$−3,5,13,21,… is an arithmetic progression with a common difference of $8$8. On the other hand, the progression $1,10,100,1000,\ldots$1,10,100,1000,… is not arithmetic because the difference between each term is not constant.
We denote the first term by the letter $a$a and the common difference by the letter $d$d. Since, $t_2=t_1+d$t2=t1+d, $t_3=t_2+d$t3=t2+d and so on, we can write any arithmetic sequence as the recurrence relation:
$t_n=t_{n-1}+d,t_1=a$tn=tn−1+d,t1=a
If we consider the recurrence relation $t_n=t_{n-1}+2,t_1=5$tn=tn−1+2,t1=5, this recurrence relation starts with $5$5 and we add $2$2 to find the next term, therefore the sequence is $5,7,9,11,13,15...$5,7,9,11,13,15...
We can also find an explicit formula in terms of $a$a and $d$d, this is useful for finding the $n$nth term without listing the sequence.
Let's look at a table of a concrete example to see the pattern for the explicit formula. For the sequence $-3,5,13,21,\ldots$−3,5,13,21,…, we have starting term of $-3$−3 and 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 | $t_n$tn | Pattern |
---|---|---|
$1$1 | $-3$−3 | $-3$−3 |
$2$2 | $5$5 | $-3+8$−3+8 |
$3$3 | $13$13 | $-3+2\times8$−3+2×8 |
$4$4 | $21$21 | $-3+3\times8$−3+3×8 |
... | ||
$n$n | $t_n$tn | $-3+(n-1)\times8$−3+(n−1)×8 |
The pattern starts to become clear and we could guess that the tenth term becomes $t_{10}=69=-3+9\times8$t10=69=−3+9×8 and the one-hundredth term $t_{100}=789=-3+99\times8$t100=789=−3+99×8. And following the pattern, the explicit formula for the $n$nth term is $t_n=-3+(n-1)\times8$tn=−3+(n−1)×8.
We could create a similar table for the arithmetic progression with starting value $a$a and common difference $d$d and we would observe the same pattern. Hence, generating the explicit rule for any arithmetic sequence is given by:
$t_n=a+\left(n-1\right)d$tn=a+(n−1)d
For any arithmetic sequence with starting value $a$a and common difference $d$d, we can express it in either of the following two forms:
$t_n=t_{n-1}+d$tn=tn−1+d, where $t_1=a$t1=a
$t_n=a+\left(n-1\right)d$tn=a+(n−1)d
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: $t_n=a+(n-1)d$tn=a+(n−1)d
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: $t_n=87+\left(n-1\right)\times\left(-7\right)$tn=87+(n−1)×(−7) or $t_n=87-7(n-1)$tn=87−7(n−1).
Hence, the $30$30th term is:$t_n=87+\left(30-1\right)\times\left(-7\right)$tn=87+(30−1)×(−7)$t_{30}=87-7\times29=-116$t30=87−7×29=−116.
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 $t_n$tn and rearrange for $n$n. Finding $n$nis determining which term position has a value of 186.
Do: This is an arithmetic sequence with $a=10$a=10 and common difference $d=4$d=4. Hence, the general rule is: $t_n=10+\left(n-1\right)\times4$tn=10+(n−1)×4, we can simplify this to $t_n=6+4n$tn=6+4n, by expanding brackets and collecting like terms. Substituting $t_n=186$tn=186, we get:
$186$186 | $=$= | $6+4n$6+4n |
$\therefore4n$∴4n | $=$= | $180$180 |
$n$n | $=$= | $45$45 |
Hence, the $45$45th term in the sequence is $186$186.
If an arithmetic sequence has $T_5=38$T5=38 and $T_9=66$T9=66, find the recurrence relation for the sequence.
Think: Consider how many times the common difference would need to be added to get from $T_5$T5 to $T_9$T9.
$T_5$T5 | $T_6$T6 | $T_7$T7 | $T_8$T8 | $T_9$T9 | ||||
$+d$+d | $+d$+d | $+d$+d | $+d$+d |
Do: The common difference needs to be added to $T_5$T5 four times to get to $T_9$T9. Therefore $T_5+4d=T_9$T5+4d=T9 so $38+4d=66$38+4d=66, this can be rearranged to solve for $d$d. Alternatively, we can use logic, an increase from $38$38 to $66$66 is made up of $4$4 common differences, so $d=(66-38)/4$d=(66−38)/4.
The common difference found to be $7$7, then we know that, using equation $a+4\times7=38$a+4×7=38 and so $a$a is $10$10. The recurrence relation for this sequence is given by:
$T_n=T_{n-1}+7,T_1=10$Tn=Tn−1+7,T1=10
The $n$nth term in an arithmetic progression is given by the formula $T_n=15+5\left(n-1\right)$Tn=15+5(n−1).
Determine $a$a, the first term in the arithmetic progression.
Determine $d$d, the common difference.
Determine $T_9$T9, the $9$9th term in the sequence.
The first term of an arithmetic sequence is $2$2. The fifth term is $26$26.
Solve for $d$d, the common difference of the sequence.
Write a recursive rule for $T_n$Tn in terms of $T_{n-1}$Tn−1 which defines this sequence and an initial condition for $T_1$T1.
Write both parts on the same line separated by a comma.
An arithmetic progression has a first term of $T_1=a$T1=a and a common difference of $d$d.
Two of the terms in the sequence are $T_7=43$T7=43 and $T_{14}=85$T14=85.
Determine $d$d, the common difference.
Determine $a$a, the first term in the sequence.
State the equation for $T_n$Tn, the $n$nth term in the sequence.
Hence find $T_{25}$T25, the $25$25th term in the sequence.
For any arithmetic sequence in the general form given by $t_n=a+\left(n-1\right)d$tn=a+(n−1)d, we can expand the bracket and collect like terms, creating a new generating rule of the form $t_n=dn+k$tn=dn+k where $d$d and $k$k are constants. For example, the rule $t_n=5+\left(n-1\right)\times2$tn=5+(n−1)×2 is equivalent to $t_n=2n+3$tn=2n+3. This 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, they will all lie on a straight line with the gradient being the common difference. This makes sense since we have a constant rate of change.
The first term is represented by the point shown at $n=1$n=1, $t_1=5$t1=5 and we can see the gradient here is the common difference $d=2$d=2.
We could also be expected to recognise an arithmetic sequence from a table, such as:
$n$n | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 |
---|---|---|---|---|---|
$t_n$tn | $5$5 | $7$7 | $9$9 | $11$11 | $13$13 |
Here, we can read off the initial term $t_1=5$t1=5 and the common difference can be seen in step between the $t_n$tn values in the second row.
This interactive tool can show us how arithmetic sequences are actually linear relationships.
The $n$nth term of an arithmetic progression is given by the equation $T_n=12+4\left(n-1\right)$Tn=12+4(n−1).
Complete the table of values.
$n$n | $1$1 | $2$2 | $3$3 | $4$4 | $10$10 |
---|---|---|---|---|---|
$T_n$Tn | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
By how much are consecutive terms in the sequence increasing?
Plot the points in the table on the graph.
If the points on the graph were joined, they would form:
a straight line
a curved line
The plotted points represent terms in an arithmetic sequence:
Complete the table of values for the given points.
$n$n | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|
$T_n$Tn | $\editable{}$ | $\editable{}$ | $\editable{}$ | $\editable{}$ |
Identify $d$d, the common difference between consecutive terms.
Write a simplified expression for the general $n$nth term of the sequence, $T_n$Tn.
Find the $15$15th term of the sequence.
The given table of values represents terms in an arithmetic sequence.
$n$n | $1$1 | $2$2 | $3$3 | $4$4 |
---|---|---|---|---|
$T_n$Tn | $9$9 | $17$17 | $25$25 | $33$33 |
Identify $d$d, the common difference between consecutive terms.
Write a simplified expression for the general $n$nth term of the sequence, $T_n$Tn.
Find the $15$15th term of the sequence.