Answer:
Recursive:
[tex]f(1)=4, f(n)=f(n-1)-4[/tex]
Explicit:
[tex]f(n)=4-4(n-1)[/tex]
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:
[tex]f(1)=a , f(n)=f(n-1)+(d)[/tex]
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:
[tex]f(1)=4, f(n)=f(n-1)-4[/tex]
Explicit Rule:
The standard format for an explicit rule for an arithmetic sequence is:
[tex]f(n)=a+d(n-1)[/tex]
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:
[tex]f(n)=4-4(n-1)[/tex]
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:
[tex]f(20)=4-4(20-1)[/tex]
Evaluate:
[tex]\begin{aligned} f(20)&=4-4(19) \\ f(20)&=4-76 \\ f(20)&=-72 \end{aligned}[/tex]
Hence, our 20th term is -72.
