The following is a geometric sequence 5,3,1,-1

No it is not.  All geometric sequences have a common ratio that is a constant found when you divide any term by the previous one.

3/5 !=1/3

So it is not a geometric sequence.

It IS an arithmetic sequence though, as all arithmetic sequences have a common difference that is a constant found when you find the difference between any term and the term preceding it.

3-5=1-3=-1-1=d=2

So there is a  common difference of 2 so this is an arithmetic sequence.  And all arithmetic sequences can be expressed as:

a(n)=a+d(n-1), a(n)=nth term, a=initial term, d=common difference, n=term number, in this case a=5 and d=-2 so

a(n)=5-2(n-1)  which of course can be simplified...

a(n)=5-2n+2

a(n)=7-2n