Given n, no of elements in the series, find the summation of the series 1, 3, 6, 10….n. The series mainly represents triangular numbers.

Examples:

Input: 2 Output: 4 Explanation: 1 + 3 = 4 Input: 4 Output: 20 Explanation: 1 + 3 + 6 + 10 = 20

A **simple solution** is to one by one add triangular numbers.

## C++

`/* CPP program to find sum` ` ` `series 1, 3, 6, 10, 15, 21...` `and then find its sum*/` `#include <iostream>` `using` `namespace` `std;` `// Function to find the sum of series` `int` `seriesSum(` `int` `n)` `{` ` ` `int` `sum = 0;` ` ` `for` `(` `int` `i=1; i<=n; i++)` ` ` `sum += i*(i+1)/2;` ` ` `return` `sum;` `}` `// Driver code` `int` `main()` `{` ` ` `int` `n = 4;` ` ` `cout << seriesSum(n);` ` ` `return` `0;` `}` |

## Java

`// Java program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum*/` `import` `java.io.*;` `class` `GFG {` ` ` ` ` `// Function to find the sum of series` ` ` `static` `int` `seriesSum(` `int` `n)` ` ` `{` ` ` `int` `sum = ` `0` `;` ` ` `for` `(` `int` `i = ` `1` `; i <= n; i++)` ` ` `sum += i * (i + ` `1` `) / ` `2` `;` ` ` `return` `sum;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main (String[] args)` ` ` `{` ` ` `int` `n = ` `4` `;` ` ` `System.out.println(seriesSum(n));` ` ` ` ` `}` `}` `// This article is contributed by vt_m` |

## Python3

`# Python3 program to find sum` `# series 1, 3, 6, 10, 15, 21...` `# and then find its sum.` `# Function to find the sum of series` `def` `seriessum(n):` ` ` ` ` `sum` `=` `0` ` ` `for` `i ` `in` `range` `(` `1` `, n ` `+` `1` `):` ` ` `sum` `+` `=` `i ` `*` `(i ` `+` `1` `) ` `/` `2` ` ` `return` `sum` ` ` `# Driver code` `n ` `=` `4` `print` `(seriessum(n))` `# This code is Contributed by Azkia Anam.` |

## C#

`// C# program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum*/` `using` `System;` `class` `GFG {` ` ` `// Function to find the sum of series` ` ` `static` `int` `seriesSum(` `int` `n)` ` ` `{` ` ` `int` `sum = 0;` ` ` ` ` `for` `(` `int` `i = 1; i <= n; i++)` ` ` `sum += i * (i + 1) / 2;` ` ` ` ` `return` `sum;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` `int` `n = 4;` ` ` ` ` `Console.WriteLine(seriesSum(n));` ` ` `}` `}` `// This article is contributed by vt_m.` |

## PHP

`<?php` `// PHP program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum` `// Function to find` `// the sum of series` `function` `seriesSum(` `$n` `)` `{` ` ` `$sum` `= 0;` ` ` `for` `(` `$i` `= 1; ` `$i` `<= ` `$n` `; ` `$i` `++)` ` ` `$sum` `+= ` `$i` `* (` `$i` `+ 1) / 2;` ` ` `return` `$sum` `;` `}` `// Driver code` `$n` `= 4;` `echo` `(seriesSum(` `$n` `));` `// This code is contributed by Ajit.` `?>` |

## Javascript

`<script>` `// javascript program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum` `// Function to find the sum of series` `function` `seriesSum(n)` `{` ` ` `let sum = 0;` ` ` `for` `(let i = 1; i <= n; i++)` ` ` `sum += i * ((i + 1) / 2);` ` ` `return` `sum;` `}` `// Driver code` `let n = 4;` ` ` `document.write(seriesSum(n)) ;` `// This code is contributed by aashish1995` `</script>` |

Output:

20

Time complexity : O(n)

An **efficient solution **is to use direct formula n(n+1)(n+2)/6

Let g(i) be i-th triangular number. g(1) = 1 g(2) = 3 g(3) = 6 g(n) = n(n+1)/2

Let f(n) be the sum of the triangular numbers 1 through n. f(n) = g(1) + g(2) + ... + g(n) Then: f(n) = n(n+1)(n+2)/6

How can we prove this? We can prove it by induction. That is, prove two things :

- It’s true for some n (n = 1, in this case).
- If it’s true for n, then it’s true for n+1.

This allows us to conclude that it’s true for all n >= 1.

Now 1) is easy. We know that f(1) = g(1) = 1. So it's true for n = 1. Now for 2). Suppose it's true for n. Consider f(n+1). We have: f(n+1) = g(1) + g(2) + ... + g(n) + g(n+1) = f(n) + g(n+1) Using our assumption f(n) = n(n+1)(n+2)/6 and g(n+1) = (n+1)(n+2)/2, we have: f(n+1) = n(n+1)(n+2)/6 + (n+1)(n+2)/2 = n(n+1)(n+2)/6 + 3(n+1)(n+2)/6 = (n+1)(n+2)(n+3)/6 Therefore, f(n) = n(n+1)(n+2)/6

Below is the implementation of the above approach:

## C++

`/* CPP program to find sum` ` ` `series 1, 3, 6, 10, 15, 21...` `and then find its sum*/` `#include <iostream>` `using` `namespace` `std;` `// Function to find the sum of series` `int` `seriesSum(` `int` `n)` `{` ` ` `return` `(n * (n + 1) * (n + 2)) / 6;` `}` `// Driver code` `int` `main()` `{` ` ` `int` `n = 4;` ` ` `cout << seriesSum(n);` ` ` `return` `0;` `}` |

## Java

`// java program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum` `import` `java.io.*;` `class` `GFG` `{` ` ` `// Function to find the sum of series` ` ` `static` `int` `seriesSum(` `int` `n)` ` ` `{` ` ` `return` `(n * (n + ` `1` `) * (n + ` `2` `)) / ` `6` `;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main (String[] args) {` ` ` ` ` `int` `n = ` `4` `;` ` ` `System.out.println( seriesSum(n));` ` ` ` ` `}` `}` `// This article is contributed by vt_m` |

## Python3

`# Python 3 program to find sum` `# series 1, 3, 6, 10, 15, 21...` `# and then find its sum*/` `# Function to find the sum of series` `def` `seriesSum(n):` ` ` `return` `int` `((n ` `*` `(n ` `+` `1` `) ` `*` `(n ` `+` `2` `)) ` `/` `6` `)` `# Driver code` `n ` `=` `4` `print` `(seriesSum(n))` `# This code is contributed by Smitha.` |

## C#

`// C# program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum` `using` `System;` `class` `GFG {` ` ` ` ` `// Function to find the sum of series` ` ` `static` `int` `seriesSum(` `int` `n)` ` ` `{` ` ` `return` `(n * (n + 1) * (n + 2)) / 6;` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` `int` `n = 4;` ` ` ` ` `Console.WriteLine(seriesSum(n));` ` ` `}` `}` `// This code is contributed by vt_m.` |

## PHP

`<?php` `// PHP program to find sum` `// series 1, 3, 6, 10, 15, 21...` `// and then find its sum` `// Function to find` `// the sum of series` `function` `seriesSum(` `$n` `)` `{` ` ` `return` `(` `$n` `* (` `$n` `+ 1) *` ` ` `(` `$n` `+ 2)) / 6;` `}` `// Driver code` `$n` `= 4;` `echo` `(seriesSum(` `$n` `));` `// This code is contributed by Ajit.` `?>` |

## Javascript

`<script>` `/* javascript program to find sum` ` ` `series 1, 3, 6, 10, 15, 21...` `and then find its sum*/` `// Function to find the sum of series` `function` `seriesSum( n)` `{` ` ` `return` `(n * (n + 1) * (n + 2)) / 6;` `}` `// Driver code` ` ` `let n = 4;` ` ` `document.write(seriesSum(n));` `// This code is contributed by todaysgaurav` `</script>` |

Output:

20

Time complexity : O(1)

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