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The sequence $u_{n}$ is defined by

\begin{aligned} u_{1} & = 3 \\ u_{n + 1} & = a u_{n} + b \end{aligned}

Given that $u_{2} = 1$ and $u_{3} = 6,$ find the values of $a$ and $b .$

By the recursive formula given, we have

\begin{aligned} u_{2} & = a u_{1} + b \\ = 3 a + b \end{aligned}

Since the value of $u_{2}$ is given to us as $1,$ we can say that $3 a + b = 1.$

Can you do something similar with $u_{3}$ to find another equation relating $a$ and $b$ ?

You will need to use simultaneous equations to solve for $a$ and $b .$