Question

finally, what is the general term definition for this sequence? 2, 12, 72, 432, ... recursive definition: \

$$\begin{cases} f(0) = 2 \\\\ f(n) = 6f(n - 1) \\end{cases}$$

general term definition: f(n) = ?(\quad)^n

Explanation:

Step1: Identify the sequence type

The recursive formula \( f(n) = 6f(n - 1) \) with \( f(0)=2 \) indicates a geometric sequence. In a geometric sequence, the general term is \( f(n)=a\times r^{n} \), where \( a \) is the initial term and \( r \) is the common ratio.

Step2: Determine \( a \) and \( r \)

• The initial term \( a \) (when \( n = 0 \)) is given by \( f(0)=2 \), so \( a = 2 \).
• The common ratio \( r \) can be found from the recursive formula \( f(n)=6f(n - 1) \), so \( r = 6 \).

Answer:

\( f(n)=2\times(6)^{n} \) (So the first box is \( 2 \) and the second box is \( 6 \))