Answer:
[tex]f(n)=-6n+36[/tex]
Step-by-step explanation:
The sequence is arithmetic because it is going down/up by the same number each time.
It is going down by 6.
This tells us that the common difference is -6.
The first term is 30.
The following is the explicit form of an arithmetic sequence:
[tex]f(n)=f(1)+d(n-1)[/tex]
where [tex]f(1)[/tex] is the first term and [tex]d[/tex] is the common difference.
So we have [tex]f(1)=30[/tex] and [tex]d=-6[/tex].
Plugging this information into our explicit formula:
[tex]f(n)=30+-6(n-1)[/tex]
[tex]f(n)=30-6n+6[/tex]
[tex]f(n)=30+6-6n[/tex]
[tex]f(n)=36-6n[/tex]
[tex]f(n)=-6n+36[/tex]
Let's check it.
The first term is [tex]f(1)=30[/tex]:
[tex]f(1)=-6(1)+36=-6+36=30[/tex] which is what the pattern reads for the first term.
The second term is [tex]f(2)=24[/tex]:
[tex]f(1)=-6(2)+36=-12+36=24[/tex] which is what the pattern reads for the second term.
The third term is [tex]f(3)=18[/tex]:
[tex]f(3)=-6(3)+36=-18+36=18[/tex] which is what the pattern reads for the third term.
The fourth term is [tex]f(4)=12[/tex]:
[tex]f(4)=-6(4)+36=-24+36=12[/tex] which is what the pattern reads for the fourth term.
This rule certainly does generate the terms of our sequence as we have shown above.
The explicit form with no doubt is [tex]f(n)=-6n+36[/tex].
