Question

find a general term definition for this sequence.\

$$\begin{cases}a_0 = -3 \\\\ a_n = a_{n - 1} + 4\\end{cases}$$

\\(a_n = ?n + \quad\\)

Explanation:

Step1: Identify the type of sequence

This is an arithmetic sequence since each term is obtained by adding a constant difference (4) to the previous term. The general form of an arithmetic sequence is \( a_n = dn + c \), where \( d \) is the common difference and \( c \) is the initial term when \( n = 0 \).

Step2: Determine the common difference \( d \)

Looking at the recurrence relation \( a_n = a_{n - 1} + 4 \), the common difference \( d \) is 4. So the coefficient of \( n \) in the general term will be 4.

Step3: Determine the constant term \( c \)

When \( n = 0 \), \( a_0 = -3 \). In the general form \( a_n = dn + c \), substituting \( n = 0 \) gives \( a_0 = d(0) + c = c \). Since \( a_0 = -3 \), then \( c = -3 \).

Answer:

\( a_n = 4n + (-3) \) (or simply \( a_n = 4n - 3 \))