Let's break down the process of writing a recursive equation to model the given arithmetic sequence [tex]\( a_n = -4 + 1.5(n - 1) \)[/tex].
### Step 1: Identify the First Term
The first term in an arithmetic sequence, denoted [tex]\( a_1 \)[/tex], can be found by substituting [tex]\( n = 1 \)[/tex] into the given formula.
[tex]\[
a_1 = -4 + 1.5(1 - 1)
\][/tex]
Calculate the expression inside the parentheses first:
[tex]\[
1 - 1 = 0
\][/tex]
Then multiply by 1.5:
[tex]\[
1.5 \times 0 = 0
\][/tex]
Thus,
[tex]\[
a_1 = -4 + 0 = -4
\][/tex]
So, the first term [tex]\( a_1 \)[/tex] is [tex]\(-4\)[/tex].
### Step 2: Identify the Common Difference
The common difference in an arithmetic sequence is the coefficient of [tex]\( n \)[/tex] in the term formula. From the sequence formula [tex]\( a_n = -4 + 1.5(n -1) \)[/tex], the coefficient of [tex]\( (n - 1) \)[/tex] is 1.5.
Thus, the common difference [tex]\( d \)[/tex] is 1.5.
### Step 3: Write the Recursive Equation
The general form of a recursive equation for an arithmetic sequence is:
[tex]\[
a_n = a_{n-1} + d
\][/tex]
Here, [tex]\( a_{n-1} \)[/tex] represents the previous term, and [tex]\( d \)[/tex] is the common difference. Given our values:
- The first term [tex]\( a_1 = -4 \)[/tex]
- The common difference [tex]\( d = 1.5 \)[/tex]
The recursive equation is therefore:
[tex]\[
a_n = a_{n-1} + 1.5 \quad \text{with} \quad a_1 = -4
\][/tex]
This equation means that each term in the sequence is obtained by adding 1.5 to the previous term, starting from the first term which is [tex]\(-4\)[/tex].
### Summary
Chris can write the recursive equation to model the same arithmetic sequence as follows:
[tex]\[
a_1 = -4
\][/tex]
[tex]\[
a_n = a_{n-1} + 1.5 \quad \text{for} \quad n \geq 2
\][/tex]
This step-by-step breakdown allows us to convert the explicit formula [tex]\( a_n = -4 + 1.5(n - 1) \)[/tex] into a recursive equation suitable for modeling the same arithmetic sequence.
