Question

what is the recursive definition for this sequence? 13, 9, 5, 1, ... \\(\

$$\begin{cases} f(0) = 13 \\\\ f(n) = ?f(n - 1) + \\square \\end{cases}$$

\\)

Explanation:

Step1: Analyze the sequence difference

The sequence is 13, 9, 5, 1,... The difference between consecutive terms: \(9 - 13=- 4\), \(5 - 9=-4\), \(1 - 5=-4\). So each term is previous term minus 4.

Step2: Express the recursive formula

A linear recursive formula is \(f(n)=a\times f(n - 1)+b\). Since we are subtracting 4, we can write it as \(f(n)=1\times f(n - 1)-4\) (because multiplying by 1 doesn't change the term, then subtracting 4 gives the next term).

Answer:

The first box (coefficient of \(f(n - 1)\)) is \(1\) and the second box (the constant term) is \(-4\). So the recursive definition is \(

$$\begin{cases}f(0) = 13\\f(n)=1\times f(n - 1)-4\end{cases}$$

\)