The numbers of a sequence are called terms where the $n$th term of a sequence is denoted by the symbol $t_n$.

## The Equation

Arithmetic sequences are defined by the following equation: $t_n=t_n-1 +d$, where $d$ is constant.

To find the terms of an arithmetic sequence, the following equation is used: $t_n=a+(n-1)d$

Where $t_n$ is the $n$th term $a$ is the first term $d$ is the common difference

### Example 1

Find the 10th term of the following arithmetic sequence $15,11,7,3...$ $a=15$ $n=10$ $d=t_2-t_1$ $d=11-15$ $d=-4$ $t_n=a+(n-1)d$ $t_{10}=15+(10-1)(-4)$ $t_{10}=15+(-36)$ $t_{10}=-21$

Given that $a=15$ and $d=-4$, the 10th term is -21.

If the first three terms of an arithmetic sequence are 11,19 and 27, which term is equal to 115? $a=11$ $d=t_2-t_1$ $d=19-11$ $d=8$

Use the rule for the arithmetic sequence: $t_n=a+(n-1)d$ $t_n=11+(n-1)8$ $t_n=11+8n-8$ $t_n=8n+3$

Now that the rule has been simplified, to find the term equal to 115, substitute the value of $t_n$ into the equation. $115=8n+3$ $n=\frac {112}{8}$ $n=14$