We want to write a recursive rule for the given sequence. To do that, we first need to identify whether the given sequence is arithmetic, geometric, or neither. To do so we will calculate the difference and ratio between consecutive terms.

We can see above that neither the ratios nor the differences are the same. Therefore, the sequence is neither geometric nor arithmetic, so we have to look for another pattern. To do so we need to consider how the consecutive terms are related. Let's take a look at the quotients between pairs of consecutive terms.

a_{1} a_{2} \frac{a _{3}}{a _{2}} \frac{a _{4}}{a _{3}} \frac{a _{5}}{a _{4}} \frac{a _{6}}{a _{5}} ⋮ = = = = = = 12 \frac{2}{2} \frac{4}{2} \frac{8}{4} \frac{3 2}{8} ⋮ = = = = 1224 ⋮

$\frac{\frac{1}{1}}{\frac{1}{1}} \frac{a _{n}}{a _{n - 1}} = a_{n - 2} \frac{\frac{1}{1}}{\frac{1}{1}}$

We can see above that the quotient between the consecutive terms equals

a_{n - 2} .

Therefore, to obtain the value of the term in the

n^{th}

position, we need to multiply two previous terms

a_{n - 1}

and

a_{n - 2} .

With this information and knowing that the first and second term equal

1

and

2

we can write the recursive formula.

a_{1} = 1, a_{2} = 2, a_{n} = a_{n - 1} \cdot a_{n - 2}

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