The sequence we are given is:
3, 5, 9, 17
and we are asked to find if it is arithmetic, geometric or neither.
To check for arithmetic we observe if there is a common difference between consecutive terms:
a(n+1) - a(n)
5-3 = 2
9-5 = 4
17-9 = 8
So clearly this is NOT an arithmetic sequence
To check for geometric sequence, we study if there is a "common ratio" by looking at the quotient of consecutive terms:
a(n+1) / a(n)
5/3
9/5
17/9
These are all different, therefore this is NOT a geometric sequence either.
Then, we answer NEITHER.
In order to find the term a(n), we notice how we build the different terms:
5 = 3 + 2
9 = 5 + 2^2
17 = 9 + 2^3
so we can try to write the a(n+1) term in terms of the previous one a(n) as:
a(n+1) = a(n) + 2^(n)
this is called a RECURRENT definition of the sequence.