Tetrahedral Numbers

A number is termed as a tetrahedral number if it can be represented as a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number is the sum of the first n triangular numbers.

The first ten tetrahedral numbers are:
1, 4, 10, 20, 35, 56, 84, 120, 165, 220, …

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Formula for nth tetrahedral number:

```Tn = (n * (n + 1) * (n + 2)) / 6
```

Proof:

```The proof uses the fact that the nth tetrahedral
number is given by,

Trin = (n * (n + 1)) / 2

It proceeds by induction.

Base Case
T1 = 1 = 1 * 2 * 3 / 6

Inductive Step
Tn+1 = Tn + Trin+1

Tn+1 = [((n * (n + 1) * (n + 2)) / 6] + [((n + 1) * (n + 2)) / 2]

Tn+1 = (n * (n + 1) * (n + 2)) / 6
```

Below is the implementation of above idea :

C++

 `// CPP Program to find the ` `// nth tetrahedral number ` `#include ` `using` `namespace` `std; ` ` `  `int` `tetrahedralNumber(``int` `n) ` `{ ` `    ``return` `(n * (n + 1) * (n + 2)) / 6; ` `} ` ` `  `// Driver Code ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `     `  `    ``cout << tetrahedralNumber(n) << endl; ` ` `  `    ``return` `0; ` `} `

Java

 `// Java Program to find the ` `// nth tetrahedral number ` `class` `GFG { ` `     `  `// Function to find Tetrahedral Number ` `static` `int` `tetrahedralNumber(``int` `n) ` `{ ` `    ``return` `(n * (n + ``1``) * (n + ``2``)) / ``6``; ` `} ` ` `  `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` `    ``int` `n = ``5``; ` `     `  `    ``System.out.println(tetrahedralNumber(n)); ` `} ` `} ` ` `  `// This code is contributed by Manish Kumar Rai. `

Python

 `# Python3 Program to find the ` `# nth tetrahedral number ` ` `  `def` `tetrahedralNumber(n): ` `     `  `    ``return` `(n ``*` `(n ``+` `1``) ``*` `(n ``+` `2``)) ``/` `6` ` `  `# Driver Code ` `n ``=` `5` `print` `(tetrahedralNumber(n)) `

C#

 `// C# Program to find the ` `// nth tetrahedral number ` `using` `System; ` ` `  `public` `class` `GFG{ ` `     `  `    ``// Function to find Tetrahedral Number ` `    ``static` `int` `tetrahedralNumber(``int` `n) ` `    ``{ ` `        ``return` `(n * (n + 1) * (n + 2)) / 6; ` `    ``} ` `     `  `    ``// Driver code ` `    ``static` `public` `void` `Main () ` `    ``{ ` `        ``int` `n = 5; ` `     `  `        ``Console.WriteLine(tetrahedralNumber(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by Ajit. `

PHP

 ` `

Output:

```35
```

Time Complexity: O(1).

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Improved By : Manish_100, jit_t, Mithun Kumar