Compute the formulas of the differences and differences in differences for the formula that make up Figure 1.23.
That is, compute the forumula that predicts the differences between the pyramidal numbers and the differences between the differences.
for Squares, it goes something like:
delta = F(N+1) - F(N) = (X+1)^2 - x^2
= X^2 + 2x + 1 - x^2
= 2x + 1
delta delta = delta x+1 - delta x
= 2(x+1) + 1 - (2x + 1)
= 2x + 2 + 1 - 2x -1
= 2x - 2x + 2 + 1 -1
Thus the table in Figure 1.23 is correct for squares. Do the same for F(x) = (K^2 +k)/2 [triangle] and F(x) = (3k^2 - k)/2 [pentagonal]
The point of the exercise is to see how the differences actually arise, to get practice with basic algebra.