Question

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?

Answer 1

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

`f(1)=2000+750(1)`

`f(1)=2750`


`f(2)=2000+750(2)`

`f(2)=2000+1500`

`f(2)=3500`


`f(3)=2000+750(3)`

`f(3)=2000+2250`

`f(3)=4250`

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
`d=a_(n)-a_(n-1)`; in other words we are going subtract the current number from the previous one:

`d=4250-3000=750`


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

We can conclude that the sequence the function
`f(n)` generates is an arithmetic sequence.