Write it like this:
\begin{array}{ccccc}n & f(n) & \Delta f(n) & \Delta^2 f(n) & \Delta^3 f(n) \\ 0 & 0 & 2 & 1 & 0 \\1 & 2 & 3 & 1 & 0 \\ 2 & 5 & 4 & 1 & 0\\ 3 & 9 & 5 & 1 \\ 4 & 14 & 6 \\ 5 & 20 \end{array}
The first column are the "n" values. The second column are the sequence values: f(1)= 2, f(2)= 5, f(3)= 9, f(4)= 14, and f(5)= 20, as you gave. The third column are the "first difference" values also as you gave: the difference between two consecutive values in the second column. The third column is the "second differences": the difference between two consecutive values in the second column. Because those are consecutive integers, 2, 3, 4, 5, the "third differences", the difference between two consecutive values in the third column, is always 1. All further diffences, then, will be 0 and can be ignored.
I put in the top row, n= 0, by working backwards: if 3-a= 2, then a= 2; if 2- a= 2, then a= 0.
Now, as I said before, Newton's divided difference formula says
f(n)= f(0)+ \Delta f(0) n+ \frac{\Delta^2 f(0)}{2}n(n-1)+ ...
Looking at the chart, f(0)= 0, \Delta f(0)= 2, and \Delta^2 f(0)= 1
Therefore,
f(n)= 0+ 2n+ \frac{1}{2}n(n-1)
Simplify that and check its values for n= 1, 2, 3, 4, and 5.
