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.