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The periodic sequence $u_{n}$ is defined by

\begin{aligned} u_{1} & = - 2 \\ u_{n + 1} & = \frac{u_{n} + 3}{p} \end{aligned}

Given that the period of the sequence is $2$ , find the value of $p$ .

Work out $u_{2}$ and $u_{3}$ . Because the period is $2$ , we know that $u_{3} = u_{1}$ .

You get two values for $p$ , only one of them is valid (the other choice makes the sequence have period $1$ ).