Question

what is the recursive definition for this sequence? 13, 8, 3, -2, ... \

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

Explanation:

Step1: Find \( f(0) \)

The first term of the sequence is 13, so when \( n = 0 \), \( f(0)=13 \).

Step2: Analyze the common difference

To find the pattern, calculate the difference between consecutive terms: \( 8 - 13=-5 \), \( 3 - 8 = -5 \), \( -2 - 3=-5 \). So the sequence is arithmetic with a common difference of -5. For a recursive formula of an arithmetic sequence, \( f(n)=f(n - 1)+d \), where \( d=-5 \) and the coefficient of \( f(n - 1) \) is 1. So \( f(n)=1\times f(n - 1)+(-5) \).

Answer:

\( f(0) = 13 \), \( f(n)=1\times f(n - 1)+(-5) \) (or \( f(n)=f(n - 1)-5 \))