Tetrahedral numbers, also known as triangular pyramidal numbers, are a type of figurate number that represents a pyramid with a triangular base and three sides. They are the three-dimensional analog of triangular numbers.
Formula and Representation
The nth tetrahedral number is given by the formula:
\[T_n = \frac{n(n+1)(n+2)}{6}\]
Where:
\(T_n\) is the nth tetrahedral number
\(n\) is the position in the sequence (a positive integer)
Calculation Example
Let's calculate the 5th tetrahedral number:
Use the formula: \(T_5 = \frac{5(5+1)(5+2)}{6}\)
Simplify: \(T_5 = \frac{5 \times 6 \times 7}{6}\)
Calculate: \(T_5 = \frac{210}{6} = 35\)
Visual Representation
This diagram represents the 5th tetrahedral number as a triangular pyramid with 35 items.
Properties and Applications
Tetrahedral numbers have several interesting properties and applications in mathematics and science:
They represent the number of items in a tetrahedral stack
They are related to combinations in probability theory
They appear in Pascal's triangle and binomial coefficients
They have applications in chemistry for molecular structures
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