DISCOVERY PROJECT

Finding Patterns

The ancient Greeks studied triangular numbers, square numbers, pentagonal numbers, and other polygonal numbers, like those shown in the figure.


To find a pattern for such numbers, we construct a first difference sequence by taking differences of successive terms; we repeat the process to get a second difference sequence, third difference sequence, and so on. For the sequence of triangular numbers $T_n$ we get the following difference table:


We stop at the second difference sequence because it's a constant sequence.
Assuming that this sequence will continue to have constant value $1$, we can work backward from the bottom row to find more terms of the first difference sequence, and from these, more triangular numbers.

If a sequence is given by a polynomial function and if we calculate the first differences, the second differences, the third differences, and so on, then eventually we get a constant sequence. For example, the triangular numbers are given by the polynomial $T_n = \frac 12 n^2 + \frac 12 n$ (see the margin note on the next page); the second difference sequence is the constant sequence $1,1,1, . . . $

The formula for the $n$th triangular number can be found using the formula for the sum of the first $n$ whole numbers. From the definition of $T_n$ we have $$ \begin{array}{} T_n &= 1 + 2 \;+\; . . . + \;n\\ &= \frac {n(n + 1)}{2}\\ &= \frac 12 n^2 + \frac 12 n \end{array} $$

  1. Construct a difference table for the square numbers and the pentagonal numbers. Use your table to find the tenth pentagonal number.
  2. From the patterns you've observed so far, what do you think the second difference would be for the hexagonal numbers? Use this, together with the fact that the first two hexagonal numbers are $1$ and $6$, to find the first eight hexagonal numbers.
  3. Construct difference tables for $C_n = n^3$. Which difference sequence is constant? Do the same for $F_n = n^4$.
  4. Make up a polynomial of degree $5$ and construct a difference table. Which difference sequence is constant?
  5. The first few terms of a polynomial sequence are $1, 2, 4, 8, 16, 31, 57, . . . $ Construct a difference table and use it to find four more terms of this sequence.