Figure 6 : How many cubes in cubes ?
Product no. Program 12 24
Size2 = 1 size1 + 8 size1 = 1^3 + 2^3 = 9 = 3^2
Size3 = 1^3 + 2^3 + 3^3 = 36 = 6^2
Size5 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 = 225 = 15^2
Hence Sizen = 1^3 + 2^3 +.... + n^3 = (n*(n+1)/2)^2
Note : 3^3 means 3 to power 3 and 2*3 means 2 times 3
Title .... Use cubes in cubes showing Sum[n^3]
Note ..... Where n^3 = n to power 3
Q1. Why size3 cubes have 1 size3, 8 size2 and 27 size1 >
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* 27 size1 is obivously true.
* 01 size3 is also true which is the whole block
* 08 size2 is not easy to see. We use the front surface as example
* ... left top 4 squares is the surface of 1 size2
* ... left bottom 4 squares is also a surface of 1 size2
* ... Right top 4 squares is also a surface of 1 size2
* ... Right bottom 4 squares is also the surface of 1 size2
* ... Hence we can make 4 size2 cubes
* ... Can you see that we can make total 8 size2 cubes
Q2. How to prove that 1^3 + 2^3 + 3^3 + .... + n^3 = (n*(n+1)/2)^2?
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* 1 : Charactor of the sequence is given in week 02 study
* 2 : Four methods of proof are given in week 02
Q3. What is the meaning of Product # 12 24 ?
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* It is chapter 12 and program 24 in Mathematics dictionary
* It displays various size1 size2 ... figures
* It concludes total cubes is (n*(n+1)/2)^2
* It also give as quiz : display a patter and give answer.