Mathematics Dictionary
Dr. K. G. Shih
Sequences : Cube Patterns
Symbol Defintion
Example : Sqr(x) = square root of x
Q01 |
- Numbers in cube patterns
Q02 |
- Difference of sequence of 1, 8, 27, 64, 125, ....
Q03 |
- Find S(n) = 1 + 8 + 27 + 64 + 81 + ..... + n^3
Q01. Numbers in cubic patterns
Cubes in cubes patterns
Cubes in cubes
Patterns
Examples
1. Sketch the pattern for 4^3
2. Sketch the pattern for 5^3
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Q02. Squences : 1, 8, 27, 64, 125, 36, ....
Difference
* 1st difference F(k) = T(k + 1) - T(K) : 07 19 37 61 .......
* 2nd difference G(k) = F(k + 1) - F(k) : 12 18 24 30 .......
* 3rd difference H(k) = G(k + 1) - G(k) : 06 06 06 06 .......
Formula
* nth term : T(n) = n^3
* Sum of n terms : S(n) = (n*(n + 1)/2)^2
* This is same as cubic function y = x^3
* 2nd derivative y" = 6 and it is the same as 3rd difference
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Q03 Find S(n) = 1 + 8 + 27 + 64 + 125 + ..... + n^3
Prove that S(n) = n*(n + 1)*(2*n + 1)/6
Sum[n^3]
= (n*(n + 1)/2)^2
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