What is the next number in the sequence 2,9,25,50,84,127,

Find the rates of change until it is constant...

dy/dx=7,16,25,34,43

d2y/dx2=8,9,9,9  so it appears 8 is the anomaly while the acceleration constant is 9 after the initial term.  Let's set up a system ignoring the first data point and solve for the quadratic...

16a+4b+c=50
9a+3b+c=25
4a+2b+c=9  gettting differences

7a+b=25
5a+b=16  and again

2a=9, a=4.5, making 5a+b=16 become:

22.5+b=16

b=-6.5, making 4a+2b+c=9 become:

18-13+c=9, c=4

So the sequence can be expressed as:

a(n)=4.5n^2-6.5n+4, where n=term number

The second and fourth answer are correct