Question

finally, what is the general term definition for this sequence?
-1, -5, -25, -125, ...
recursive definition:
\

$$\begin{cases} f(0) = -1 \\\\ f(n) = 5f(n - 1) \\end{cases}$$

general term definition:
$f(n) = \boxed{?}(\boxed{})^n$

Explanation:

Step1: Identify the sequence type

The sequence -1, -5, -25, -125,... is a geometric sequence. In a geometric sequence, the general term is given by \( f(n)=a\times r^{n} \), where \( a \) is the first term and \( r \) is the common ratio.

Step2: Determine the first term (\(a\))

The first term \( f(0) = - 1 \), so \( a=-1 \).

Step3: Determine the common ratio (\(r\))

To find the common ratio, we divide a term by its previous term. For example, \( \frac{-5}{-1}=5 \), \( \frac{-25}{-5} = 5 \), \( \frac{-125}{-25}=5 \). So the common ratio \( r = 5 \).

Step4: Write the general term

Using the formula for the general term of a geometric sequence \( f(n)=a\times r^{n} \), substituting \( a=-1 \) and \( r = 5 \), we get \( f(n)=-1\times(5)^{n} \) or \( f(n)=(-1)(5)^{n} \).

Answer:

\( f(n)= - 1\times(5)^{n} \) (or \( f(n)=(-1)(5)^{n} \), so the first box is -1 and the second box is 5)