The method of finite differences can sometimes be used to guess a formula f(n) (but not to prove it). The way I am about to describe it requires that the values of f(n) be given for integer values of n beginning with n=0 (rather than 1). Let's use the example of guessing the formula for the sum of the cubes of the first n positive integers. We calculate a few values:
and then arrange them in a row. Then we build a difference table beneath them by subtraction:
We continue generating rows by this process until it appears that we get a row of 0's. This does not always happen, but when it does, we can build a polynomial formula for f(n). Using the first entries in each row, multiply the first entry of the first row by 1/0!, the first entry of the second row by n/1!, the first entry of the third row by n(n-1)/2!, the first entry of the fourth row by n(n-1)(n-2)/3!, the first entry of the fifth row by n(n-1)(n-2)(n-3)/4!, etc., and then add the results. (Remember that 0! is defined to be 1.) In our example, we get
which simplifies (you try it!) to
At this point we have not proved that this formula is correct, but at least we have a good guess for the formula, and we can try to prove it, e.g., by induction.