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A R I T H M E T I C

Theoretic Arithmetic

Appendix 3  Section 2

# The sum of consecutive cubes

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When the same number is repeated as a factor three times -- as 4 × 4 × 4 -- we call the product the 3rd power of that base; that product is commonly called a cube. (This is analogous to the volume of the solid figure called a cube.)

Here is number 4:

Upon repeatedly adding it four times --

-- we have the 2nd power, or the square, of 4.

Upon repeatedly adding that power four times --

-- we have the 3rd power, or the cube, of 4.

It will be convenient for the moment to express the cube of a number with the exponent 3.

 13 = 1 23 = 8 33 = 27 43 = 64

We come now to one of the most remarkable facts in the structure of the natural numbers:

The sum of n consecutive cubes is equal to the square
of the nth triangle.

13 + 23 + 33 + . . . + n3 = (1 + 2 + 3 + . . . + n)2.

To see that, we will begin here:

The difference between the squares of two consecutive triangular numbers
is a cube.

Triangles:  1  3  6  10  15  21  28

 32 − 12 = 23 62 − 32 = 33 102 − 62 = 43 152 − 102 = 53

The base of each cube is the difference of the two triangles.

Look -- here is the cube of 4:

From it, let us separate this rectangular array --

-- and reposition it here:

Then we have the square of side 10 --

-- minus the square of side 6.

The difference between the squares of those two consecutive triangles, 10 and 6, is equal to a cube, 43.

Therefore,

The sum of those four cubes is equal to the square of the fourth triangle.

*

Alternatively, since every square number is the sum of consecutive odd numbers, so is the square of a triangular number.

 12 = 1 (1 + 2)2 = 1 + 3 + 5 = 9 (1 + 2 + 3)2 = 1 + 3 + 5 + 7 + 9 + 11 = 36 (1 + 2 + 3 + 4)2 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100

Therefore, the difference of those squares -- each cube -- will be a sum of consecutive odd numbers, although not starting with 1.

 1 = 1 9 − 1 = 8 = 3 + 5 36 − 9 = 27 = 7 + 9 + 11 100 − 36 = 64 = 13 + 15 + 17 + 19

Again, the sum of those four cubes is equal to the square of the fourth triangle.

We proved that by mathematical induction
in Topic 27 of Precalculus.

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