Other people have given calculus answers... here is one that doesn't require calculus. Its very tedious, but fairly simple to understand.
Give your DT "coefficients" work your way back to the solution.
I.e. in this case: you start with
48
48
48
From which you can get
104 <-- Note: this needs to be given!
152
200
248
And work your way back to
9
73
241
561
1081
1849
So you know you want a polynomial P(x)such that
P(1) = 9
p(2) = 73
...
p(3) = 1849
So now you know you need at most a 5th degree polynomial.
So, you have y = a_0 + a_1 * x + a_2 * x^2 + ... + a_5 * x^5
Plug in each value of x and you get 6 equations with 6 unknowns. Solve for a_0, ..., a_5.
Bunch of the a's might well be 0, but thats ok.
Edit: This only works if your difference table eventually reduces to a set of differences which are all the same (e.g here 48, 48, 48). Otherwise, the answer is not a polynomial.
The order of the polynomial is equal to the number of difference steps until a constant is reached. It MAY be as long as the number of samples in the series.
So difference tables are also a handy way to determine the order of a series!
Give your DT "coefficients" work your way back to the solution.
I.e. in this case: you start with
48
48
48
From which you can get
104 <-- Note: this needs to be given!
152
200
248
And work your way back to
9
73
241
561
1081
1849
So you know you want a polynomial P(x)such that P(1) = 9
p(2) = 73
...
p(3) = 1849
So now you know you need at most a 5th degree polynomial.
So, you have y = a_0 + a_1 * x + a_2 * x^2 + ... + a_5 * x^5
Plug in each value of x and you get 6 equations with 6 unknowns. Solve for a_0, ..., a_5.
Bunch of the a's might well be 0, but thats ok.
Edit: This only works if your difference table eventually reduces to a set of differences which are all the same (e.g here 48, 48, 48). Otherwise, the answer is not a polynomial.