# Sum of square-sums of first n natural numbers

Given a positive integer n. The task is to find the sum of the sum of square of first n natural number.

Examples :

```Input : n = 3
Output : 20
Sum of square of first natural number = 1
Sum of square of first two natural number = 1^2 + 2^2 = 5
Sum of square of first three natural number = 1^2 + 2^2 + 3^2 = 14
Sum of sum of square of first three natural number = 1 + 5 + 14 = 20

Input : n = 2
Output : 6
```

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

Method 1: O(n) The idea is to find sum of square of first i natural number, where 1 <= i <= n. And them add them all.
We can find sum of squares of first n natural number by fromula: n * (n + 1)* (2*n + 1)/6

Below is the implementation of this approach:

## C++

 `// CPP Program to find the sum of sum of  ` `// squares of first n natural number ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find sum of sum of square of  ` `// first n natural number ` `int` `findSum(``int` `n) ` `{ ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 1; i <= n; i++)  ` `        ``sum += ((i * (i + 1) * (2 * i + 1)) / 6); ` `    ``return` `sum; ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `n = 3; ` `    ``cout << findSum(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java Program to find the sum of  ` `// sum of squares of first n natural ` `// number ` ` `  `class` `GFG { ` `     `  `    ``// Function to find sum of sum of ` `    ``// square of first n natural number ` `    ``static` `int` `findSum(``int` `n) ` `    ``{ ` `        ``int` `sum = ``0``; ` `        ``for` `(``int` `i = ``1``; i <= n; i++)  ` `            ``sum += ((i * (i + ``1``)  ` `               ``* (``2` `* i + ``1``)) / ``6``); ` `        ``return` `sum; ` `    ``} ` `     `  `    ``// Driver Program ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``3``; ` `     `  `        ``System.out.println( findSum(n)); ` `    ``} ` `} ` ` `  ` `  `// This code is contributed by  ` `// Arnab Kundu `

## Python3

 `# Python3 Program to find the sum ` `# of sum of squares of first n  ` `# natural number ` ` `  `# Function to find sum of sum of ` `# square of first n natural number ` `def` `findSum(n): ` `    ``summ ``=` `0` `    ``for` `i ``in` `range``(``1``, n``+``1``): ` `        ``summ ``=` `(summ ``+` `((i ``*` `(i ``+` `1``)  ` `                ``*` `(``2` `*` `i ``+` `1``)) ``/` `6``))  ` `    ``return` `summ  ` ` `  `# Driven Program ` `n ``=` `3` `print``(``int``(findSum(n))) ` ` `  ` `  `# This code is contributed by  ` `# Prasad Kshirsagar `

## C#

 `// C# Program to find the sum of sum of  ` `// squares of first n natural number ` `using` `System; ` ` `  `public` `class` `GFG { ` `     `  `    ``// Function to find sum of sum of ` `    ``// square of first n natural number ` `    ``static` `int` `findSum(``int` `n) ` `    ``{ ` `        ``int` `sum = 0; ` `        ``for` `(``int` `i = 1; i <= n; i++)  ` `            ``sum += ((i * (i + 1) *  ` `                  ``(2 * i + 1)) / 6); ` `        ``return` `sum; ` `    ``} ` `     `  `    ``// Driver Program ` `    ``static` `public` `void` `Main() ` `    ``{ ` `        ``int` `n = 3; ` `         `  `        ``Console.WriteLine(findSum(n)); ` `    ``} ` `} ` ` `  `// This code is contributed by ` `// Arnab Kundu. `

## PHP

 ` `

Output :

```20
```

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

Method 2: O(1)
Mathematically, we need to find, Σ ((i * (i + 1) * (2*i + 1)/6), where 1 <= i <= n
So, lets solve this summation,

```Sum = Σ ((i * (i + 1) * (2*i + 1)/6), where 1 <= i <= n
= (1/6)*(Σ ((i * (i + 1) * (2*i + 1)))
= (1/6)*(Σ ((i2 + i) * (2*i + 1))
= (1/6)*(Σ ((2*i3 + 3*i2 + i))
= (1/6)*(Σ 2*i3 + Σ 3*i2 + Σ i)
= (1/6)*(2*Σ i3 + 3*Σ i2 + Σ i)
= (1/6)*(2*(i*(i + 1)/2)2 + 3*(i * (i + 1) * (2*i + 1))/6 + i * (i + 1)/2)
= (1/6)*(i * (i + 1))((i*(i + 1)/2) + ((2*i + 1)/2) + 1/2)
= (1/12) * (i * (i + 1)) * ((i*(i + 1)) + (2*i + 1) + 1)
= (1/12) * (i * (i + 1)) * ((i2 + 3 * i + 2)
= (1/12) * (i * (i + 1)) * ((i + 1) * (i + 2))
= (1/12) * (i * (i + 1)2 * (i + 2))
```

Below is the implentation of this approach:

## C++

 `// CPP Program to find the sum of sum of  ` `// squares of first n natural number ` `#include ` `using` `namespace` `std; ` ` `  `// Function to find sum of sum of square ` `// of first n natural number ` `int` `findSum(``int` `n) ` `{ ` `    ``return` `(n * (n + 1) * (n + 1) * (n + 2)) / 12; ` `} ` ` `  `// Driven Program ` `int` `main() ` `{ ` `    ``int` `n = 3; ` `    ``cout << findSum(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java Program to find the sum of sum of  ` `// squares of first n natural number ` `class` `GFG { ` ` `  `    ``// Function to find sum of sum of  ` `    ``// square of first n natural number ` `    ``static` `int` `findSum(``int` `n) ` `    ``{ ` `        ``return` `(n * (n + ``1``) * ` `        ``(n + ``1``) * (n + ``2``)) / ``12``; ` `    ``} ` `     `  `    ``// Driver Program ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``3``; ` `         `  `        ``System.out.println(findSum(n) ); ` `    ``} ` `} ` ` `  `// This code is contributed by Arnab Kundu `

## Python3

 `# Python3 Program to find the sum  ` `# of sum of squares of first n  ` `# natural number ` ` `  `# Function to find sum of sum of ` `# square of first n natural number ` `def` `findSum(n): ` `    ``return` `((n ``*` `(n ``+` `1``) ``*` `(n ``+` `1``) ` `                 ``*` `(n ``+` `2``)) ``/` `12``) ` ` `  `# Driven Program ` `n ``=` `3` `print``(``int``(findSum(n))) ` ` `  `# This code is contributed by  ` `# Prasad Kshirsagar `

## C#

 `// C# Program to find the sum of sum of  ` `// squares of first n natural number ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// Function to find sum of sum of ` `    ``// square of first n natural number ` `    ``static` `int` `findSum(``int` `n) ` `    ``{ ` `        ``return` `(n * (n + 1) * (n + 1) ` `                     ``* (n + 2)) / 12; ` `    ``} ` `     `  `    ``// Driver Program ` `    ``static` `public` `void` `Main() ` `    ``{ ` `        ``int` `n = 3; ` `         `  `        ``Console.WriteLine(findSum(n) ); ` `    ``} ` `} ` ` `  `// This code is contributed by Arnab Kundu `

## PHP

 ` `

Output:

```20
```

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

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up Check out this Author's contributed articles.

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.

Article Tags :
Practice Tags :

Be the First to upvote.

Please write to us at contribute@geeksforgeeks.org to report any issue with the above content.