Question

what is the recursive definition for this sequence? -1, 1, -1, 1, ... \

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

Explanation:

Step1: Determine the initial term

The sequence starts with -1 when \( n = 0 \), so \( f(0)=-1 \).

Step2: Find the recursive relation

Looking at the sequence: \( -1, 1, -1, 1, \dots \). To get from one term to the next, we multiply by -1. Let's check: \( f(1)=-1\times f(0)=-1\times(-1) = 1 \), \( f(2)=-1\times f(1)=-1\times1=-1 \), etc. So the recursive formula is \( f(n)=-1\times f(n - 1)+0 \) (since there's no addition, just multiplication by -1).

Answer:

\( f(0)=\boldsymbol{-1} \), \( f(n)=\boldsymbol{-1}f(n - 1)+\boldsymbol{0} \)