5,9,21,37,81,? what no. will come next? how?

Question

options 1]163
2]153
3]183
4]203
it is a series .can you solve?

Answer 1

Again,

5
9=5*2-1
21=9*2+3
37=21*2-5
81=37*2+7

see the pattern
so the next number must be
81*2-9
=153

Answer is 2) 153

Answer 2

x=[5,9,21,37,81]

using least squares, the coefficients for a fourth order polynomial fit are very round.
7/6
-37/3
293/6
-221/3
41

Those numbers are very clean.
The next number in the polynomial series is 205

mod(x-1,4) is zero for all terms in the current series.
the only one of the potential answers for which this is true is
153

Given that mod(x,6)=[5,3,3,1,3] it could be guessed that the next term in the sequence when taken mod 6 has a result of 3 making the mod series 5,3,3,1,3,3,.. If that were true then again the next term would have to be 153.