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?
Accepted Solution
A:
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]
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.