Finding the Terms of an Arithmetic Sequence

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.

$115=8n+3$
$n=\frac {112}{8}$
$n=14$

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