Question

what is the recursive definition for this sequence? -1, -5, -25, -125, ... \\(\

$$\begin{cases} f(0) = ? \\\\ f(n) = \\square f(n - 1) + \\square \\end{cases}$$

\\)

Explanation:

Step1: Find the initial term \( f(0) \)

The first term of the sequence is when \( n = 0 \), and the first term is \( - 1 \), so \( f(0)=-1 \).

Step2: Determine the common ratio

To find the recursive formula, we look at the relationship between consecutive terms. Let's check the ratio between \( f(n) \) and \( f(n - 1) \). For example, \( \frac{-5}{-1}=5 \), \( \frac{-25}{-5} = 5 \), \( \frac{-125}{-25}=5 \). So the sequence is a geometric sequence with a common ratio of \( 5 \). In the recursive formula \( f(n)=a\times f(n - 1)+b \), since it's a geometric sequence (multiplicative), \( b = 0 \) and \( a=5 \).

Answer:

\( f(0)=-1 \), \( f(n) = 5f(n - 1)+0 \)