SequencesSolving linear recurrence relations

A sequence is defined by the recurrence relation \({U_n} = m{U_{n - 1}} + c\)

Find the values of \(m\) and \(c\) if \({U_1} = - 3\), \({U_2} = 7\) and \({U_3} = 10\).

\({U_n} = m{U_{n - 1}} + c\)

First, take the recurrence relation you were given and substitute in the values given for \({U_1}\), \({U_2}\) and \({U_3}\)

\({U_2} = m{U_1} + c \Rightarrow 7\)

\(= m( - 3) + c \Rightarrow - 3m + c = 7\)

\({U_3} = m{U_2} + c \Rightarrow 10\)

\(= m(7) + c \Rightarrow 7m + c = 10\)

Now solve the equations \(- 3m + c = 7\) and \(7m + c = 10\) simultaneously.

\(- 10m = - 3\)

\(m = 0.3\)

\(7m + c = 10\). Substitute this value for \(m\) in one of the equations above to find the value of \(c\).

\(7(0.3) + c = 10\)

\(c = 10 - 2.1\)

\(c = 7.9\)

You can now use these values to find \({U_4}\) and \({U_0}\).

First, put the values of \(m\) and \(c\) into the recurrence relation, giving \({U_n} = 0.3{U_{n - 1}} + 7.9\)

Now you can use this relation to find \({U_4}\) and \({U_0}\).

To find \({U_4}\) use \({U_3}\) and to find \({U_0}\) use \({U_1}\).

\({U_4} = 0.3{U_3} + 7.9\)

\(= 0.3(10) + 7.9\)

\(= 10.9\)

\({U_1} = 0.3{U_0} + 7.9\)

\(- 3 = 0.3{U_0} + 7.9\)

\(0.3{U_0} = - 3 - 7.9\)

\(0.3{U_0} = - 10.9\)

\({U_0} = - \frac{{109}}{3}\)

\({U_4} = 10.9\) and \({U_0} = - \frac{{109}}{3}\)