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.