# n’th Pentagonal Number

• Difficulty Level : Medium
• Last Updated : 25 Jan, 2022

Given an integer n, find the nth Pentagonal number. The first three pentagonal numbers are 1, 5, and 12 (Please see below diagram).
The n’th pentagonal number Pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots when the pentagons are overlaid so that they share one vertex [Source Wiki]
Examples :

```Input: n = 1
Output: 1

Input: n = 2
Output: 5

Input: n = 3
Output: 12```

Recommended Practice

In general, a polygonal number (triangular number, square number, etc) is a number represented as dots or pebbles arranged in the shape of a regular polygon. The first few pentagonal numbers are: 1, 5, 12, etc.
If s is the number of sides in a polygon, the formula for the nth s-gonal number P (s, n) is

```nth s-gonal number P(s, n) = (s - 2)n(n-1)/2 + n

If we put s = 5, we get

n'th Pentagonal number Pn = 3*n*(n-1)/2 + n```

Examples:

Pentagonal Number Below are the implementations of the above idea in different programming languages.

## C++

 `// C++ program for above approach``#include``using` `namespace` `std;` `// Finding the nth pentagonal number``int` `pentagonalNum(``int` `n)``{``    ``return` `(3 * n * n - n) / 2;``}` `// Driver code``int` `main()``{``    ``int` `n = 10;``    ` `    ``cout << ``"10th Pentagonal Number is = "``         ``<< pentagonalNum(n);` `    ``return` `0;``}` `// This code is contributed by Code_Mech`

## C

 `// C program for above approach``#include ``#include ` `// Finding the nth Pentagonal Number``int` `pentagonalNum(``int` `n)``{``    ``return` `(3*n*n - n)/2;``}` `// Driver program to test above function``int` `main()``{``    ``int` `n = 10;``    ``printf``(``"10th Pentagonal Number is = %d \n \n"``,``                             ``pentagonalNum(n));` `    ``return` `0;``}`

## Java

 `// Java program for above approach``class` `Pentagonal``{``    ``int` `pentagonalNum(``int` `n)``    ``{``        ``return` `(``3``*n*n - n)/``2``;``    ``}``}` `public` `class` `GeeksCode``{``    ``public` `static` `void` `main(String[] args)``    ``{``        ``Pentagonal obj = ``new` `Pentagonal();``        ``int` `n = ``10``;   ``        ``System.out.printf(``"10th petagonal number is = "``                          ``+ obj.pentagonalNum(n));``    ``}``}`

## Python3

 `# Python program for finding pentagonal numbers``def` `pentagonalNum( n ):``    ``return` `(``3``*``n``*``n ``-` `n)``/``2``#Script Begins` `n ``=` `10``print` `(``"10th Pentagonal Number is = "``, pentagonalNum(n))`` ` `#Scripts Ends`

## C#

 `// C# program for above approach``using` `System;` `class` `GFG {``    ` `    ``static` `int` `pentagonalNum(``int` `n)``    ``{``        ``return` `(3 * n * n - n) / 2;``    ``}` `    ``public` `static` `void` `Main()``    ``{``        ``int` `n = 10;``        ` `        ``Console.WriteLine(``"10th petagonal"``        ``+ ``" number is = "` `+ pentagonalNum(n));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

 ``

## Javascript

 ``

Output :

`10th Pentagonal Number is = 145`

Time Complexity: O(1)
Auxiliary Space: O(1)
Reference:
https://en.wikipedia.org/wiki/Polygonal_number
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