Question

finally, what is the general term definition for this sequence? -4, -3, -2, -1, ... recursive definition: \

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

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

Explanation:

Step1: Identify sequence type

This is an arithmetic sequence with first term \( f(0) = -4 \) and common difference \( d = 1 \) (from recursive formula \( f(n)=f(n - 1)+1 \)).

Step2: 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 \( f(0)= - 4 \) and \( d = 1 \) into the formula: \( f(n)=-4 + 1\times n=n - 4 \), which can be written as \( f(n)=1\times n+(-4) \).

Answer:

For the coefficient of \( n \): \( 1 \), for the constant term: \( -4 \) (so the general term is \( f(n)=1n - 4 \) or \( f(n)=n - 4 \))