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The sequence $ℓ_{n}$ is defined by

\begin{aligned} ℓ_{1} & = 2 \\ ℓ_{2} & = 9 \\ ℓ_{n + 2} & = \frac{1 + ℓ_{n + 1}}{ℓ_{n}} \end{aligned}

Find the period of this seqence.

Compute the next few values of $ℓ_{n}$ , beginning with

ℓ_{3} = \frac{1 + ℓ_{2}}{ℓ_{1}}

Because the next term depends on the previous two terms, you can't stop once a single number gets repeated. You can only stop when a pair of consecutive terms has appeared before in the sequence.

You need to compute as far as $ℓ_{7}$ to get the period.