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=2\times 8+4

Now use t_2 to calculate t_3

t_3=2\times 20+4

Now use t_3 to find t_4

t_4=2\times 44+4

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}

Now we can use t_4 to find t_3.

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

The third term is 78.

See Also:

First Order Difference Equations
Finding the Terms of an Arithmetic Sequence
Finding the Terms of a Geometric Sequence