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