Question

What is the 105th term of the sequence: 21, 17, 13, 9

Answer 1

The first thing we are going to to is checking if the sequence is arithmetic of geometric. A sequence is arithmetic if it has a common difference, $d$. A sequence is geometric if it has a common ratio, $r$.
Lets test for a common difference first. To do that we are going to subtract the current term from the previous term:
$9-13=-4$
$13-17=-4$
$17-21=-4$
Since we have a common difference, $d=-4$, or sequence is arithmetic. 
Now, to find its 105 term, we are going to find its explicit formula using the general form of an arithmetic sequence: $a_{n}=a_{1}+(n-1)d$
where
$a_{n}$ is the nth term
$a_{1}$ is the first term
$n$ is the position of the term in the sequence 
$d$ is the common diference
We can infer for our sequence that $a_{1}=21$, and for previews calculations we know that $d=-4$. So lets replace those values:
$a_{n}=a_{1}+(n-1)d$
$a_{n}=21+(n-1)(-4)$
$a_{n}=21-4n+4$
$a_{n}=25-4n$
Finally, to find the 105th term of the sequence, we just need to replace $n$ with 105 in our explicit formula:
$a_{n}=25-4n$
$a_{105}=25-4(105)$
$a_{105}=25-420$
$a_{105})=-395$

We can conclude that the 105th term of the sequence is -395.