Generating the Terms of a Sequence Defined by a First Order Difference Equation

We can generate the terms of a sequence by using the rule for first order difference equations.

First Order Difference Equation

A reminder of what a general first order difference equation looks like:

$t_{n+1}=t_n+b$ $t_1=a$

Example 1

Find the first four terms of the sequence defined by the first order difference equation:

$t_n=2t_{n-1}+4$ $t_1=8$

We know that the starting term ( $t_1$ ) so we can generate the next term ( $t_2$ ) by completing the equation.

$t_2=2t_1+4$
$t_2=2\times 8+4$
$t_2=20$

Now use $t_2$ to calculate $t_3$

$t_3=2t_2+4$
$t_3=2\times 20+4$
$t_3=44$

Now use $t_3$ to find $t_4$

$t_4=2t_3+4$
$t_4=2\times 44+4$
$t_4=92$

The first four terms are 8, 20, 44 and 92.

Example 2

A sequence is defined by the first order difference equation:

$t_{n+1}=4t_n+6$ $n=1, 2, 3, ...$

If $t_5=1,278$ , what is the third term?

We need to calculate the previous term so transpose the equation to make $t_n$ the subject.

$t_n=\frac {t_{n+1}-6}{4}$

Now, use $t_5$ to find $t_4$ using the transposed equation.

$t_4=\frac {t_5-6}{4}$
$t_4=\frac {1,278-6}{4}$
$t_4=318$

Now we can use $t_4$ to find $t_3$ .

$t_3=\frac {t_4-6}{4}$
$t_3=\frac {318-6}{4}$
$t_3=78$

The third term is 78.

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