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Theoretic Arithmetic Appendix 3 Section 2 ## The sum of consecutive cubesWhen 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 -- 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.
We come now to one of the most remarkable facts in the structure of the natural numbers: The sum of 1 To The difference between the squares of two consecutive triangular numbers Triangles: 1 3 6 10 15 21 28
The base of each cube is the Look -- here is the cube of 4: From it, let us separate this rectangular array -- -- and reposition it here: Then we have the -- The difference between the squares of those two consecutive triangles, 10 and 6, is equal to a cube, 4 Therefore, The * Alternatively, since every square number is the sum of consecutive odd numbers, so is the square of a triangular number.
Therefore, the
Again, the
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