Recursive sequences | Mathematico

We have already seen explicit sequences, where the terms are give by a formula. Sometimes, a sequence can be defined instead by doing some operation on the previous terms. So $u_{n + 1}$ is given by doing something to $u_{n} .$ Such a sequence is called a recursive sequence.

In the next example, we will see how this works.

The sequence $u_{n}$ is defined by

\begin{aligned} u_{1} & = 5 \\ u_{n + 1} & = 2 u_{n} + 1 \end{aligned}

Find $u_{2}, u_{3}$ and $u_{4}$
Show that
$\begin{aligned} u_{3} & = 4 u_{1} + 3 \\ u_{4} & = 4 u_{2} + 3 \end{aligned}$
Prove, in general, that
$u_{n + 1} = 4 u_{n - 1} + 3$