Certainly! Let's analyze the given information and derive explicit formulas or recursive equations step by step.
### Explicit Form
Chris uses the following initial term and common difference to model an arithmetic sequence:
- First term ([tex]\(a_1\)[/tex]) = 7
- Common difference ([tex]\(d\)[/tex]) = 2
The explicit form for the [tex]\(n\)[/tex]-th term ([tex]\(a_n\)[/tex]) of an arithmetic sequence is given by the formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Using the given values:
- [tex]\(a_1 = 7\)[/tex]
- [tex]\(d = 2\)[/tex]
We will substitute these values into the formula:
[tex]\[ a_n = 7 + (n - 1) \cdot 2 \][/tex]
Let's simplify:
[tex]\[ a_n = 7 + 2(n - 1) \][/tex]
[tex]\[ a_n = 7 + 2n - 2 \][/tex]
[tex]\[ a_n = 2n + 5 \][/tex]
So, the explicit form to find the [tex]\(15^{\text{th}}\)[/tex] term would be:
[tex]\[ a_{15} = 2 \cdot 15 + 5 \][/tex]
Using the above formula:
[tex]\[ a_{15} = 30 + 5 = 35 \][/tex]
### Recursive Equation
Now, we have another sequence given by Ahmad, and it's modeled as:
[tex]\[ a_1 = -4 + 1.5(n - 1) \][/tex]
First, let's rewrite the first term:
[tex]\[ a_1 = -4 + 1.5(n - 1) \][/tex]
For an arithmetic sequence, the recursive form is generally given by:
[tex]\[ a_{n+1} = a_n + d \][/tex]
To find the common difference [tex]\(d\)[/tex], note that the explicit formula given can be simplified:
[tex]\[ a_1 = -4 + 1.5(1 - 1) \][/tex]
[tex]\[ a_1 = -4 + 0 \][/tex]
[tex]\[ a_1 = -4 \][/tex]
Next, observe the term involving [tex]\( n \)[/tex]:
[tex]\[ a_n = -4 + 1.5(n - 1) \][/tex]
This implies that:
[tex]\[ a_{n+1} = -4 + 1.5(n + 1 - 1) \][/tex]
[tex]\[ a_{n+1} = -4 + 1.5n \][/tex]
We need to find the common difference [tex]\(d\)[/tex]; since the sequence is arithmetic, the difference [tex]\(d\)[/tex] between successive terms is constant.
From the explicit form above:
- The coefficient of [tex]\(n\)[/tex] (which is 1.5) is our common difference.
Hence, the recursive formula Ahmad would use is:
For the first term:
[tex]\[ a_1 = -4 \][/tex]
For the subsequent terms:
[tex]\[ a_{n+1} = a_n + 1.5 \][/tex]
So Chris can write the recursive equation for the same sequence modeled by Ahmad as:
[tex]\[ a_1 = -4 \][/tex]
[tex]\[ a_{n+1} = a_n + 1.5 \][/tex]
