Question

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

$$\begin{cases} f(0) = 13 \\\\ f(n) = f(n - 1) - 5 \\end{cases}$$

general term definition:
f(n) = ?n + \quad

Explanation:

Step1: Identify sequence type

This is an arithmetic sequence with first term \( f(0) = 13 \) and common difference \( d=-5 \).

Step2: Recall arithmetic sequence formula

The general formula for an arithmetic sequence starting at \( n = 0 \) is \( f(n)=f(0)+d\times n \).

Step3: Substitute values

Substitute \( f(0) = 13 \) and \( d=-5 \) into the formula: \( f(n)=-5n + 13 \).

Answer:

\( f(n)= -5n + 13 \) (So the coefficient of \( n \) is \(-5\) and the constant term is \(13\))