Tetrahedral number (inglés)
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Definición Nivel 1
Tetrahedral numbers are the numbers which can be made by considering a tetrahedral pattern of beads in three dimensions.
For example: if we make a triangle of beads with 3 beads to a side, and on top of this we place a triangle with two beads to a side, and on top of that a triangle with one bead to a side, we have made a tetrahedron of beads. In this case the total number of beads is (3rd triangular number)+(2nd triangular number)+(1st triangular number) =6+3+1=10.
In general the nth tetrahedral number is equal to the sum of the first n triangular numbers. This is the same as the 4th number from the left in the (n+3)th row of Pascal's triangle.
We can use the binomial formula for numbers in Pascal's triangle to show that the nth tetrahedral number is
or (n+2)(n+1)n/6.
The only numbers which are both tetrahedral and square are 4
For example: if we make a triangle of beads with 3 beads to a side, and on top of this we place a triangle with two beads to a side, and on top of that a triangle with one bead to a side, we have made a tetrahedron of beads. In this case the total number of beads is (3rd triangular number)+(2nd triangular number)+(1st triangular number) =6+3+1=10.
In general the nth tetrahedral number is equal to the sum of the first n triangular numbers. This is the same as the 4th number from the left in the (n+3)th row of Pascal's triangle.
We can use the binomial formula for numbers in Pascal's triangle to show that the nth tetrahedral number is
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The only numbers which are both tetrahedral and square are 4
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Relaciones
- Más general:
- (en) Integer
- Referencia:
- (en) Choose
- (en) Pascal's triangle
- (en) Triangular number
- Referenciado en:
- (en) Twenty
Financiado por: Socrates Minerva UE, HeyMath!, Cambridge University Press