The function below models the number of bonus points earned for completing the nth level of a certain video game. f(n)=2,000+750n , where n=1,  2,  3,  4,... Which sequence does the function f(n) generate?

To check what kid of sequence the function [tex]f(n)[/tex] generates, we are going to evaluate the function at [tex]n=1[/tex], [tex]n=2[/tex], and [tex]n=3[/tex] to find the first three terms of our sequence. Then, we are going to check if the sequence has a constant difference, [tex]d[/tex], which means the sequence will be arithmetic, or a constant ratio [tex]r[/tex], which means the sequence will be geometric.
[tex]f(1)=2000+750(1)[/tex]
[tex]f(1)=2750[/tex]

[tex]f(2)=2000+750(2)[/tex]
[tex]f(2)=2000+1500[/tex]
[tex]f(2)=3500[/tex]

[tex]f(3)=2000+750(3)[/tex]
[tex]f(3)=2000+2250[/tex]
[tex]f(3)=4250[/tex]

So, we have the sequence 2750,3500,4250,... Lets find if it has a constant difference. To do that we are going to use the formula [tex]d=a_{n}-a_{n-1}[/tex]; in other words we are going subtract the current number from the previous one:
[tex]d=4250-3000=750[/tex]

[tex]d=3500-2750=750[/tex]
Notice that the difference, [tex]d[/tex], is constant, so we have a arithmetic sequence.

We can conclude that the sequence the function [tex]f(n)[/tex] generates is an arithmetic sequence.