Answer:
Recursive:
![f(1)=4, f(n)=f(n-1)-4](https://tex.z-dn.net/?f=f%281%29%3D4%2C%20f%28n%29%3Df%28n-1%29-4)
Explicit:
![f(n)=4-4(n-1)](https://tex.z-dn.net/?f=f%28n%29%3D4-4%28n-1%29)
And the 20th term f(20) is -72.
Step-by-step explanation:
We have the sequence:
4, 0, -4, -8...
Let’s find the recursive and explicit rule for the sequence.
Recursive Rule:
Let’s determine the factor by which the sequence is decreasing by. We see that each subsequent term is 4 less than the previous term. In other words, our common difference is -4.
So, this is a arithmetic sequence.
The standard format for an recursive rule for an arithmetic sequence is:
![f(1)=a , f(n)=f(n-1)+(d)](https://tex.z-dn.net/?f=f%281%29%3Da%20%2C%20f%28n%29%3Df%28n-1%29%2B%28d%29)
Where a is the initial term and d is our common difference.
From our sequence, we know that our initial term a is 4.
We also determined that our common difference is -4.
Substitute. Hence, our recursive rule is:
![f(1)=4, f(n)=f(n-1)-4](https://tex.z-dn.net/?f=f%281%29%3D4%2C%20f%28n%29%3Df%28n-1%29-4)
Explicit Rule:
The standard format for an explicit rule for an arithmetic sequence is:
![f(n)=a+d(n-1)](https://tex.z-dn.net/?f=f%28n%29%3Da%2Bd%28n-1%29)
Where a is the initial term and d is the common difference. So, again, let’s substitute 4 for a and -4 for d. Hence, our explicit formula is:
![f(n)=4-4(n-1)](https://tex.z-dn.net/?f=f%28n%29%3D4-4%28n-1%29)
To find f(20), we can use the explicit formula. It is possible to use the recursive formula, but it gets tedious. Therefore, we will substitute 20 for n for our explicit formula:
![f(20)=4-4(20-1)](https://tex.z-dn.net/?f=f%2820%29%3D4-4%2820-1%29)
Evaluate:
![\begin{aligned} f(20)&=4-4(19) \\ f(20)&=4-76 \\ f(20)&=-72 \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20f%2820%29%26%3D4-4%2819%29%20%5C%5C%20f%2820%29%26%3D4-76%20%5C%5C%20f%2820%29%26%3D-72%20%5Cend%7Baligned%7D)
Hence, our 20th term is -72.