Pay close attention to how the consecutive terms are related.
Recursive Rule: a_1 = 1, a_2 = 3, a_n=a_(n-1) * a_(n-2)
Next two terms: 243, 6561
Practice makes perfect
We want to write a recursive formula for the given sequence.
1, 3, 3, 9, 27 ...
To do so we need to analyze how the consecutive terms are related. Let's find the product of each pair of consecutive terms.
\begin{array}{ccccc}
{\color{#0000FF}{a_2}} \cdot a_1&=& {\color{#0000FF}{3}} \cdot 1 &=& {\color{#009600}{3}} \\[0.3em]
{\color{#009600}{a_3}} \cdot \col{a_2}&=& {\color{#009600}{3}} \cdot {\color{#0000FF}{3}} &=& {\color{#FF0000}{9}} \\[0.3em]
{\color{#FF0000}{a_4}} \cdot {\color{#009600}{a_3}}&=& {\color{#FF0000}{9}} \cdot {\color{#009600}{3}}&=& {\color{#FF00FF}{27}} \\[0.3em]
\vdots & & \vdots & & \vdots
\end{array}
a_(n-1) * a_(n-2) = a_n
We can see above that the product of the consecutive terms equals the next term. Therefore, to obtain the value of the term in the nth position, we need to multiply two previous terms. With this information and knowing that the first term equals 1 and the second term equals 3, we can write the recursive formula.
a_1=1, a_2 = 3 and a_n=a_(n-1) * a_(n-2)
We will write the next two terms of a sequence now.
ccccc
... & & ... & & ... [0.3em]
a_5 * a_4&=& 27 * 9 &=& 243 [0.3em]
a_6 * a_5&=& 243 * 27 &=& 6561
