Linear recurrence relations
Remember that a recurrence relation is a sequence that gives you a connection between two consecutive terms. These two terms are usually \({U_{n + 1}}\) and \({U_n}\). However they could be given as \({U_n}\) and \({U_{n - 1}}\)
Example
A sequence is given by the recurrence relation \({U_{n + 1}} = 3{U_n} + 9\).
If \({U_0} = - 4\) find the first \(5\) terms.
Given \({U_{n + 1}} = 3{U_n} + 9\) you can work out the following:
Step 1:
When \(n = 0\), \({U_0} = - 4\)
\({U_{0 + 1}} = 3{U_0} + 9\)
\(\Rightarrow {U_1} = 3( - 4) + 9 = - 3\)
\(\Rightarrow {U_1} = - 3\)
Step 2:
When \(n = 1\), \({U_1} = - 3\)
\({U_{1 + 1}} = 3{U_1} + 9\)
\(\Rightarrow {U_2} = 3( - 3) + 9 = 0\)
\(\Rightarrow {U_2} = 0\)
Step 3:
When \(n = 2\), \({U_2} = 0\)
\({U_{2 + 1}} = 3{U_2} + 9\)
\(\Rightarrow {U_3} = 3(0) + 9 = 9\)
\(\Rightarrow {U_3} = 9\)
Step 4:
When \(n = 3\), \({U_3} = 9\)
\({U_{3 + 1}} = 3{U_3} + 9\)
\(\Rightarrow {U_4} = 3(9) + 9 = 36\)
\(\Rightarrow {U_4} = 36\)
So the first 5 terms are \(- 4, - 3,0,9,36\)
