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 $\left(u_{n+1}u_n\right)$(un+1−un).
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 $u_1$u1 and the common difference by the letter $d$d. Note that $t_n$tn or $T_n$Tn may also be used instead of $u_n$un.
We can find an explicit formula in terms of $u_1$u1 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 $u_1=3$u1=−3 and $d=8$d=8. A table of the sequence is show below:
$n$n  $u_n$un  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  $3+(n1)\times8$−3+(n−1)×8 
The pattern starts to become clear and we could guess that the tenth term becomes $u_{10}=69=3+9\times8$u10=69=−3+9×8 and the onehundredth term $u_{100}=789=3+99\times8$u100=789=−3+99×8. And following the pattern, the explicit formula for the $n$nth term is $u_n=3+(n1)\times8$un=−3+(n−1)×8.
We could create a similar table for the arithmetic progression with starting value $u_1$u1 and common difference $d$d and we would observe the same pattern. Hence, generating the explicit rule for any arithmetic sequence is given by:
$u_n=u_1+\left(n1\right)d$un=u1+(n−1)d
For any arithmetic sequence with starting value $u_1$u1 and common difference $d$d, we can express it in the explicit form: $u_n=u_1+\left(n1\right)d$un=u1+(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 $u_1$u1 and common difference $d$d and substitute these into the general form: $u_n=a+(n1)d$un=a+(n−1)d
Do: Each term is a decrease from the last by $7$7. So we have an arithmetic sequence with: $u_1=87$u1=87 and $d=7$d=−7. The general formula for this sequence is: $u_n=87+\left(n1\right)\times\left(7\right)$un=87+(n−1)×(−7) or $u_n=877(n1)$un=87−7(n−1).
Hence, the $30$30th term is:
$u_n$un  $=$=  $877\left(n1\right)$87−7(n−1) 
Write down the rule. 
$\therefore\ u_{30}$∴ u30  $=$=  $877\left(301\right)$87−7(30−1) 
Substitute in $n=30$n=30. 
$=$=  $877\times29$87−7×29 
Simplify. 

$=$=  $116$−116 
Evaluate. 
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. Finding $n$n is determining which term position has a value of $186$186.
Do: This is an arithmetic sequence with $u_1=10$u1=10 and common difference $d=4$d=4. Hence, the general rule is: $u_n=10+\left(n1\right)\times4$un=10+(n−1)×4, we can simplify this to $u_n=6+4n$un=6+4n, by expanding brackets and collecting like terms. Substituting $u_n=186$un=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.
Determine which of the following sequences is an arithmetic progression.
$3,0,3,6,\ldots$3,0,−3,−6,…
$1,2,3,5,8,13,\ldots$1,2,3,5,8,13,…
$3,3^3,3^6,3^9,\ldots$3,33,36,39,…
$4,4,4,4,\ldots$4,−4,4,−4,…
$3,0,3,6,\ldots$3,0,−3,−6,…
$1,2,3,5,8,13,\ldots$1,2,3,5,8,13,…
$3,3^3,3^6,3^9,\ldots$3,33,36,39,…
$4,4,4,4,\ldots$4,−4,4,−4,…
What is the common difference of this progression?
$3,0,3,6,\ldots$3,0,−3,−6,…
The $n$nth term in an arithmetic progression is given by the formula $T_n=15+5\left(n1\right)$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.
In an arithmetic progression where $a$a is the first term, and $d$d is the common difference, $T_7=43$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.
Recognise arithmetic progressions.
Use the formulae for the nth term and for the sum of the first n terms to solve problems involving arithmetic progressions.