Question

Which recursive formula can be used to define this sequence for n>1? 2, 12, 72, 432, 2592, 15552, ...

Answer 1

To find a recursive formula for the given sequence \(2, 12, 72, 432, 2592, 15552, \ldots\), we can start by analyzing the relationship between consecutive terms.

Let's denote the sequence as \(a_n\), where:
- \(a_1 = 2\)
- \(a_2 = 12\)
- \(a_3 = 72\)
- \(a_4 = 432\)
- \(a_5 = 2592\)
- \(a_6 = 15552\)

Now, we can compute the ratios of consecutive terms:

\[
\frac{a_2}{a_1} = \frac{12}{2} = 6
\]
\[
\frac{a_3}{a_2} = \frac{72}{12} = 6
\]
\[
\frac{a_4}{a_3} = \frac{432}{72} = 6
\]
\[
\frac{a_5}{a_4} = \frac{2592}{432} = 6
\]
\[
\frac{a_6}{a_5} = \frac{15552}{2592} = 6
\]

From this analysis, we can see that each term is obtained by multiplying the previous term by 6. Therefore, we can express the recursive formula as follows:

\[
a_n = 6 \cdot a_{n-1} \quad \text{for } n > 1
\]

We also need to state the initial condition:
\[
a_1 = 2
\]

Thus, the complete recursive formula to define the sequence is:

\[
\begin{cases}
a_1 = 2 \\
a_n = 6 \cdot a_{n-1} \quad \text{for } n > 1
\end{cases}
\]