Cubes
The sum of the first n odd numbers equals n2. Because the nth odd number is 2n – 1, we can write it as follows, 1 + 3 + 5 + 7 + … + 2n – 1 = n2. It is not so well known that by grouping odd numbers and adding numbers in each group we may also get all cubes. [See Cajori, 1896, 1914, pp. 32-33; Reid, 2006, pp. 115-116; and see also Nicomachus’s theorem, http://mathworld.wolfram.com/NicomachussTheorem.html].
| 1, | 3 + 5, | 7 + 9 + 11, | 13 + 15 + 17 + 19, | 21 + 23 + 25 + 27 + 29, ... | |
| 1 | 8 | 27 | 64 | 125 |
An explanation of why it is so is algebraic and not arithmetic.
We start by showing how to represent a cube x3 of any real number x as the difference of two squares, x3 = ((x2 + x)/2)2 – ((x2 – x)/2)2.
Derivation
((x2 + x)/2)2 – ((x2 – x)/2)2 = (x4 + 2x3 +x2)/4 - (x4 - 2x3 +x2)/4 = x3
Notice that the x4’s and x2’s cancel and 2x3/4 - -2x3/4 = x3.
Remark
The equality shown above is true for all real numbers, and even for some other algebraic systems (for example, matrices), so we call it "algebraic:. Theorems which are valid only for whole numbers are called "number theoretical" or "arithmetic". So the statement that n2 is the sum of the first n odd numbers is an arithmetic theorem.
Table of values:
| x: | (x2 - x)/2: | (x2 + x)/2: | Computation of x3: |
| 1 | 0 | 1 | 12 - 02 = 13 |
| 2 | 1 | 3 | 32 - 12 = 23 |
| 3 | 3 | 6 | 62 - 32 = 33 |
| 4 | 6 | 10 | 102 - 62 = 43 |
| ... | ... | ... | ...... |
But when x is a whole number, then ((x2 + x)/2)2 is the sum of the first (x2 + x)/2 odd numbers, and ((x2 - x)/2)2 is the sum of the first (x2 – x)/2 odd numbers. Therefore x3, which is their difference, is the sum of x = (x2 – x)/2 - (x2 – x)/2, consecutive odd numbers.
For example
43 = 102 – 62 = (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19) – (1 + 3 + 5 + 7 + 9 + 11)
= 13 + 15 + 17 + 19
What we have shown also tells us that the sum of the cubes of the whole numbers from 1 to x, 13 + 23 + … + x3, equals ((x2 + x)/2)2, which is Nicomachus’s theorem, because
(x2 + x)/2 = x(x + 1)/2 = 1 + 2 + … + x.
References
Cajori, F. (1896, 1914). A history of elementary mathematics with hints on methods of teaching. New York: Macmillan.
Reid, C. (2006). From zero to infinity: What makes numbers interesting. Wellesley, MA: A.K. Peters, Ltd.