Number Theory and Algebra Working GroupTuesday, July 10"Everything is the thing of the sum of up and over" A. Cuoco Recall that if f is a function from W to W (from the counting numbers to the counting numbers), then f determines a sequences of “differences” Delta (n) where Delta (n) = f (n+1) – f (n). For example, here’s a partial table of a function. Values of a certain function f(n)
We determined that the function is f (n) = 3n+1 (assuming that Delta (n) remains constantly 5). Here’s another example. Observe that Delta Delta is the “second difference”, the sequence which record s the differences of the differences, if you will.
Last night, Gary proved the following: Theorem If
f is a function from W to W such that the second differences are constant, then f is a
quadratic function, i.e. f (n) = ax^2 + bx + c, for some real numbers a, b, c. Gary provided a proof during the meeting today. Let’s look at the function g(n) again. Observe that Delta (n) is a linear function (since the first differences for Delta (n) are constant. In fact, Delta (n) = 3n +1.
So how do we determine g (n) from Delta (n)? Observe that g (1) = g (0) + Delta (0), g (2) = g(1) + Delta(1) = g(0) + Delta(0) + Delta(1). Hey, see a pattern developing here? In general, g (n) = g (0) + Delta (0) + Delta (1) +.,,+ Delta(n-1). In Gary’s proof, the main idea was to show that if Delta (n) is linear, then the function g (n) is quadratic. Amazingly we were able to show that from the first line of a table for a function h (x) with a constant second difference, if the first line is
(where r,s,t are numbers) then h(n) = r + sn + t(n(n-1)/2) = (t/2)n^2 + (s- t/2)n + r We then discussed summation notation, how Pascal’s Triangle is lurking in the background, We had further discussion concerning binomial coefficients, the binomial theorem- we will discuss these ideas further. “I thought I did but now I don’t” John |