Question

what is the recursive definition for this sequence? 12, 20, 28, 36, ... \

$$\begin{cases} f(0) = 12 \\\\ f(n) = ?f(n - 1) + \\square \\end{cases}$$

Explanation:

Step1: Analyze the sequence difference

First, find the difference between consecutive terms. \(20 - 12 = 8\), \(28 - 20 = 8\), \(36 - 28 = 8\). So the sequence is arithmetic with a common difference of 8.

Step2: Determine the recursive formula

For an arithmetic sequence, the recursive formula is \(f(n)=f(n - 1)+d\), where \(d\) is the common difference. Here, the coefficient of \(f(n - 1)\) should be 1 (since we are just adding the common difference to the previous term), and the added term is 8. Also, the initial term \(f(0) = 12\). So \(f(n)=1\times f(n - 1)+8\).

Answer:

The first box (coefficient of \(f(n - 1)\)) is \(1\) and the second box (the added term) is \(8\). So the recursive definition is \(

$$\begin{cases}f(0) = 12\\f(n)=1\times f(n - 1)+8\end{cases}$$

\)