Answer:
[tex]f(1)=8, f(n+1)=f(n)-3[/tex]
Step-by-step explanation:
The given choices are
The given sequence is 8, 5, 2, -1,...
Where the first term is 8, the difference is -3, because the sequence is decreasing.
The arithmetic sequence is defined as
[tex]a_{n}=a_{1}+(n-1)d[/tex]
Where [tex]a_{1}=8[/tex] and [tex]d=-3[/tex]. So for a general term, the sequence is defined as
[tex]a_{n}=8+(n-1)(-3) \\a_{n}=8-3n+3\\a_{n}=11-3n[/tex]
However, notice that the given choices are using another notation, which is an easier notation actually.
[tex]f(1)=8[/tex] refers to the first term of the sequence.
We know that the difference is -3, that is, the sequence is made by adding -3 to the first term.
[tex]f(n)[/tex] is the n-term and [tex]f(n+1)[/tex] is the follwoing term.
So, notice that to find the following term [tex]f(n+1)[/tex], we just need to add -3 to the first term [tex]f(1)=8[/tex].
Therefore, the function that best defines the sequence is
[tex]f(1)=8, f(n+1)=f(n)-3[/tex]
So, the right answer is the last choice.
