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# Sum of the series 1.2.3 + 2.3.4 + … + n(n+1)(n+2)

Find the sum up to n terms of the series: 1.2.3 + 2.3.4 + … + n(n+1)(n+2). In this 1.2.3 represent the first term and 2.3.4 represent the second term .

Examples :

Input : 2
Output : 30
Explanation: 1.2.3 + 2.3.4 = 6 + 24 = 30

Input : 3
Output : 90

Simple Approach We run a loop for i = 1 to n, and find the sum of (i)*(i+1)*(i+2).
And at the end display the sum .

## C++

 `// CPP program to find sum of the series``// 1.2.3 + 2.3.4 + 3.4.5 + ...``#include ``using` `namespace` `std;` `int` `sumofseries(``int` `n)``{``    ``int` `res = 0;``    ``for` `(``int` `i = 1; i <= n; i++)``        ``res += (i) * (i + 1) * (i + 2);   ``    ``return` `res;``}` `// Driver Code``int` `main()``{``    ``cout << sumofseries(3) << endl;``    ``return` `0;``}`

## Java

 `// Java program to find sum of the series``// 1.2.3 + 2.3.4 + 3.4.5 + ...``import` `java.io.*;``import` `java.math.*;` `class` `GFG``{` `    ``static` `int` `sumofseries(``int` `n)``    ``{``    ``int` `res = ``0``;``    ``for` `(``int` `i = ``1``; i <= n; i++)``        ``res += (i) * (i + ``1``) * (i + ``2``);``    ``return` `res;``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``System.out.println(sumofseries(``3``));``    ``}``}`

## Python3

 `# Python 3 program to find sum of the series``# 1.2.3 + 2.3.4 + 3.4.5 + ...` `def` `sumofseries(n):` `    ``res ``=` `0``    ``for` `i ``in` `range``(``1``, n``+``1``):``        ``res ``+``=` `(i) ``*` `(i ``+` `1``) ``*` `(i ``+` `2``)``    ``return` `res` `# Driver Program``print``(sumofseries(``3``))` `# This code is contributed``# by Smitha Dinesh Semwal`

## C#

 `// Java program to find sum of the series``// 1.2.3 + 2.3.4 + 3.4.5 + ...``using` `System;` `class` `GFG``{` `    ``static` `int` `sumofseries(``int` `n)``    ``{``        ``int` `res = 0;``        ``for` `(``int` `i = 1; i <= n; i++)``            ``res += (i) * (i + 1) * (i + 2);``        ``return` `res;``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main()``    ``{``        ``Console.WriteLine(sumofseries(3));``    ``}``}` `// This code is contributed by vt_m.`

## PHP

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## Javascript

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Output :

`90`

Time Complexity: O(N)
Auxiliary Space: O(1)

## Efficient Approach

Using Efficient Approach we know that we have to find = summation of( (n)*(n+1)*(n+2) )

Sn = summation[ (n)*(n+1)*(n+2) ]
Sn = summation [n3 + 2*n2 + n2 + 2*n]
We know sum of cubes of natural numbers is (n*(n+1))/2)2, sum of squares of natural numbers is n * (n + 1) * (2n + 1) / 6 and sum of first n natural numbers is n(n+1)/2
Sn = ((n*(n+1))/2)2 + 3((n)*(n+1)*(2*n+1)/6) + 2*((n)*(n+1)/2)
So by evaluating the above we get,
Sn = (n*(n+1)*(n+2)*(n+3)/4)
Hence it has a O(1) complexity.

## C++

 `// Efficient CPP program to``// find sum of the series``// 1.2.3 + 2.3.4 + 3.4.5 + ...``#include ``using` `namespace` `std;` `// function to calculate``// sum of series``int` `sumofseries(``int` `n)``{``    ``return` `(n * (n + 1) *``           ``(n + 2) * (n + 3) / 4);``}` `// Driver Code``int` `main()``{``    ``cout << sumofseries(3) << endl;``    ``return` `0;``}`

## Java

 `// Efficient Java program to``// find sum of the series``// 1.2.3 + 2.3.4 + 3.4.5 + ..``import` `java.io.*;``import` `java.math.*;` `class` `GFG``{``    ``static` `int` `sumofseries(``int` `n)``    ``{``    ``return` `(n * (n + ``1``) *``           ``(n + ``2``) * (n + ``3``) / ``4``);``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``System.out.println(sumofseries(``3``));``    ``}``}`

## Python3

 `# Efficient CPP program to find sum of the``# series 1.2.3 + 2.3.4 + 3.4.5 + ...` `# function to calculate sum of series``def` `sumofseries(n):` `    ``return` `int``(n ``*` `(n ``+` `1``) ``*` `(n ``+` `2``) ``*` `(n ``+` `3``) ``/` `4``)`  `# Driver program``print``(sumofseries(``3``))``    `  `# This code is contributed``# by Smitha Dinesh Semwal`

## C#

 `// Efficient C# program to``// find sum of the series``// 1.2.3 + 2.3.4 + 3.4.5 + ..``using` `System;` `class` `GFG``{``    ``static` `int` `sumofseries(``int` `n)``    ``{``    ``return` `(n * (n + 1) *``           ``(n + 2) * (n + 3) / 4);``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main()``    ``{``        ``Console.WriteLine(sumofseries(3));``    ``}``}` `// This code is contributed by anuj_67.`

## PHP

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## Javascript

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Output :

`90`

Time Complexity: O(1)
Auxiliary Space: O(1)