Question

what is the recursive definition for this sequence? 3, 5, 7, 9, ... \

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

Explanation:

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

The sequence starts with 3 when \( n = 0 \) (assuming the first term is \( f(0) \)), so \( f(0)=3 \).

Step2: Analyze the pattern for \( f(n) \)

Looking at the sequence \( 3, 5, 7, 9, \dots \), each term is the previous term plus 2. So the coefficient of \( f(n - 1) \) is 1 (since we add the previous term as is) and the constant added is 2. So \( f(n)=1\times f(n - 1)+2 \).

Answer:

\( f(0) = 3 \), \( f(n) = 1 \cdot f(n - 1) + 2 \)