The Calculus of Finite Differences, sometimes called the method of vanishing triangles, is a simple procedure for determining which polynomial function f(n) will fit a sequence of given numerical data of the form "f(0), f(1), f(2), ...". The approach to constructing the triangle is just the reverse of that for Pascal's Triangle, that is, instead of adding subsequent entries in a row, you now subtract consecutive terms. An example will make my point clear. Let f(n) = 2n2 + 3n + 4. The sequence of values for n = 0, 1, 2, 3, ... is 4, 9, 18, 31, ... . The vanishing triangle is 4 9 18 31 5 9 13 4 4 0 (Any zeros obtained are superfluous and can be omitted in practice. They are only mentioned to illustrate how the term "triangle" is involved.) If one does several functions like the example (i.e. 2nd- degree functions), it will be noticed that: 1) the value in the 3rd row is always a constant and twice the coefficient of n2, 2) the first entry in the 2nd row equals the sum of the coefficients of n2 and n, and 3) the first entry in the first row is the constant term in f(n). Hence, if at least the first three values of f(n), beginning with f(0), are known, the coefficients can be determined by examining the first entries in the three rows.
That the foregoing is true is easily proved. Let f(n) = an2 + bn + c. Therefore, the triangle formed with the first four values is c a+b+c 4a+2b+c 9a+3b+c a+b 3a+b 5a+b 2a 2a The expressions "2a, a+b, and c" confirm the earlier statements as true. This will give the student who likes to discover formulas for himself a tool to do so. For example, what is the general formula for the triangular numbers? The sequence is 1, 3, 6, 10, ... . Though this function begins with the value for n = 1, once the triangle is set up, it's a simple matter to "back it up" one value. 0 1 3 6 10 1 2 3 4 1 1 1 As 2a = 1, a + b = 1, and c = 0, the coefficients are a = b = ½ and c = 0. Therefore, f(n) = ½ n2 + n + 0 = ½ n(n + 1) The utility of this process will be seen as one continues working with numerical functions, especially those of higher degree. For example, for data requiring a 3rd-degree function (i.e. f(n) = an3 + bn2 + cn + d), the necessary expressions are "6a, 6a + 2b, a + b + c, and d." Determining the "guide" expressions for higher degree functions just requires careful work in the triangle. This is only a brief examination into this topic, as I prefer to leave any further discoveries up to you, the reader.
This originally appeared in the Bulletin of the Kansas Association of Teachers of Matematics. February 1971, p.20.
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