Question

Which recursive formula can be used to define this sequence for n>1? 4, 36, 324, 2916, 26244, 236196, ...

Answer 1

To identify the recursive formula for the given sequence \(4, 36, 324, 2916, 26244, 236196, \ldots\), we start by examining the relationship between consecutive terms.

Let's denote the nth term of the sequence by \( a_n \):
- \( a_1 = 4 \)
- \( a_2 = 36 \)
- \( a_3 = 324 \)
- \( a_4 = 2916 \)
- \( a_5 = 26244 \)
- \( a_6 = 236196 \)

To explore a potential recursive relationship, we can divide each term by its preceding term:

\[
\frac{a_2}{a_1} = \frac{36}{4} = 9
\]
\[
\frac{a_3}{a_2} = \frac{324}{36} = 9
\]
\[
\frac{a_4}{a_3} = \frac{2916}{324} = 9
\]
\[
\frac{a_5}{a_4} = \frac{26244}{2916} = 9
\]
\[
\frac{a_6}{a_5} = \frac{236196}{26244} = 9
\]

Notice that each term is multiplied by 9 to obtain the next term in the sequence. Therefore, we can formulate the recursive relationship as:

\[
a_n = 9 \times a_{n-1}
\]

for \( n > 1 \).

To fully define the recursive rule, we can state that:

\[
a_1 = 4
\]
\[
a_n = 9 \times a_{n-1} \quad \text{for } n > 1
\]

Thus, the recursive formula is:

\[
\begin{align*}
a_1 & = 4 \\
a_n & = 9 \times a_{n-1}, \quad \text{for } n > 1.
\end{align*}
\]