# Sum of the series 2 + (2+4) + (2+4+6) + (2+4+6+8) + …… + (2+4+6+8+….+2n)

Given a positive integer n. The problem is to find the sum of the given series 2 + (2+4) + (2+4+6) + (2+4+6+8) + …… + (2+4+6+8+….+2n), where i-th term in the series is the sum of first i even natural numbers.

Examples:

```Input : n = 2
Output : 8
(2) + (2+4) = 8

Input : n = 5
Output : 70
(2) + (2+4) + (2+4+6) + (2+4+6+8) + (2+4+6+8+10) = 70
```

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

Naive Approach: Using two loops get the sum of each i-th term and then add those sum to the final sum.

## C++

 `// C++ implementation to find the sum ` `// of the given series ` `#include ` `  `  `using` `namespace` `std; ` `  `  `// function to find the sum ` `// of the given series ` `int` `sumOfTheSeries(``int` `n) ` `{ ` `    ``int` `sum = 0; ` `    ``for` `(``int` `i = 1; i <= n; i++) { ` `  `  `        ``// first term of each i-th term ` `        ``int` `k = 2; ` `        ``for` `(``int` `j = 1; j <= i; j++) { ` `            ``sum += k; ` `  `  `            ``// next term ` `            ``k += 2; ` `        ``} ` `    ``} ` `  `  `    ``// required sum ` `    ``return` `sum; ` `} ` `  `  `// Driver program to test above ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `    ``cout << ``"Sum = "` `         ``<< sumOfTheSeries(n); ` `    ``return` `0; ` `} `

## Java

 `// Java implementation to find the ` `// sum of the given series ` `class` `GFG{ ` ` `  `    ``// function to find the sum ` `    ``// of the given series ` `    ``static` `int` `sumOfTheSeries(``int` `n) ` `    ``{ ` `        ``int` `sum = ``0``; ` `        ``for` `(``int` `i = ``1``; i <= n; i++) { ` `     `  `            ``// first term of each i-th term ` `            ``int` `k = ``2``; ` `            ``for` `(``int` `j = ``1``; j <= i; j++) { ` `                ``sum += k; ` `     `  `                ``// next term ` `                ``k += ``2``; ` `            ``} ` `        ``} ` `     `  `        ``// required sum ` `        ``return` `sum; ` `    ``} ` `     `  `    ``// Driver program to test above ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``5``; ` ` `  `        ``System.out.printf(``"Sum = %d"``, ` `                     ``sumOfTheSeries(n)); ` `    ``} ` `} ` ` `  `// This code is contriubted by ` `// Smitha Dinesh Semwal `

## Python3

 `# Python3 implementation to find  ` `# the sum of the given series ` ` `  `# function to find the sum ` `# of the given series ` `def` `sumOfTheSeries(n): ` ` `  `    ``sum` `=` `0` `    ``for` `i ``in` `range``(``0``, n ``+` `1``): ` `        ``# first term of each i-th ` `        ``# term ` `        ``k ``=` `2` `        ``for` `j ``in` `range``(``1``, i ``+` `1``): ` `            ``sum` `=` `sum` `+` `k; ` ` `  `            ``# next term ` `            ``k ``=` `k ``+` `2` `     `  `    ``# required sum ` `    ``return` `sum``; ` ` `  `# Driver program to test above ` `n ``=` `5` `ans ``=` `sumOfTheSeries(n); ` `print` `(ans) ` ` `  `# This code is contributed by saloni1297. `

## C#

 `// C# implementation to find the ` `// sum of the given series ` `using` `System; ` ` `  `class` `GFG{ ` ` `  `    ``// function to find the sum ` `    ``// of the given series ` `    ``static` `int` `sumOfTheSeries(``int` `n) ` `    ``{ ` `        ``int` `sum = 0; ` `        ``for` `(``int` `i = 1; i <= n; i++) { ` `     `  `            ``// first term of each i-th term ` `            ``int` `k = 2; ` `            ``for` `(``int` `j = 1; j <= i; j++) { ` `                ``sum += k; ` `     `  `                ``// next term ` `                ``k += 2; ` `            ``} ` `        ``} ` `     `  `        ``// required sum ` `        ``return` `sum; ` `    ``} ` `     `  `    ``// Driver program to test above ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 5; ` ` `  `        ``Console.Write(``"Sum = "``+ ` `                    ``sumOfTheSeries(n)); ` `    ``} ` `} ` ` `  `// This code is contriubted by ` `// vt_m `

## PHP

 ` `

Output:

```Sum = 70
```

Efficient Approach:
Let an be the n-th term of the given series.

```an = (2 + 4 + 6 + 8 +....+ 2n).
= sum of first n even numbers.
= n * (n + 1).
= n2 + n.
```

Refer this post for the proof of above formula.

Now, Refer this and this post for the proof of above formula.

## C++

 `// C++ implementation to find the sum ` `// of the given series ` `#include ` `using` `namespace` `std; ` `  `  `// functionn to find the sum ` `// of the given series ` `int` `sumOfTheSeries(``int` `n) ` `{ ` `    ``// sum of 1st n natural numbers ` `    ``int` `sum_n = (n * (n + 1) / 2); ` `     `  `    ``// sum of squares of 1st n natural numbers ` `    ``int` `sum_sq_n = (n * (n + 1) / 2) * ` `                      ``(2 * n + 1) / 3; ` `                   `  `    ``// required sum ` `    ``return` `(sum_n + sum_sq_n); ` `} ` `  `  `// Driver program to test above ` `int` `main() ` `{ ` `    ``int` `n = 5; ` `    ``cout << ``"Sum = "` `         ``<< sumOfTheSeries(n); ` `    ``return` `0; ` `} `

## Java

 `// Java implementation to find the ` `// sum of the given series ` `class` `GFG{ ` `     `  `    ``// functionn to find the sum ` `    ``// of the given series ` `    ``static` `int` `sumOfTheSeries(``int` `n) ` `    ``{ ` `         `  `        ``// sum of 1st n natural numbers ` `        ``int` `sum_n = (n * (n + ``1``) / ``2``); ` `         `  `        ``// sum of squares of 1st n natural ` `        ``// numbers ` `        ``int` `sum_sq_n = (n * (n + ``1``) / ``2``) * ` `                        ``(``2` `* n + ``1``) / ``3``; ` `                     `  `        ``// required sum ` `        ``return` `(sum_n + sum_sq_n); ` `    ``} ` `     `  `    ``// Driver program to test above ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `n = ``5``; ` `         `  `        ``System.out.printf(``"Sum = %d"``, ` `                    ``sumOfTheSeries(n)); ` `    ``} ` `} ` ` `  `// This code is contriubted by  ` `//Smitha Dinesh Semwal `

## Python3

 `# Python3 implementation to find  ` `# the sum of the given series ` ` `  `# functionn to find the sum ` `# of the given series ` `def` `sumOfTheSeries(n): ` ` `  `    ``# sum of 1st n natural numbers ` `    ``sum_n ``=` `int``((n ``*` `(n ``+` `1``) ``/` `2``)); ` `     `  `    ``# sum of squares of 1st n natural numbers ` `    ``sum_sq_n ``=` `int` `((n ``*` `(n ``+` `1``) ``/` `2``) ``*` `(``2` `*` `n ``+` `1``) ``/` `3``) ` `                     `  `    ``# required sum ` `    ``return` `(sum_n ``+` `sum_sq_n); ` ` `  `# Driver program to test above ` `n ``=` `5` `ans ``=` `sumOfTheSeries(n) ` `print` `(ans) ` ` `  `# This code is contributed by saloni1297. `

## C#

 `// C# implementation to find the ` `// sum of the given series ` `using` `System; ` ` `  `class` `GFG{ ` `     `  `    ``// functionn to find the sum ` `    ``// of the given series ` `    ``static` `int` `sumOfTheSeries(``int` `n) ` `    ``{ ` `         `  `        ``// sum of 1st n natural numbers ` `        ``int` `sum_n = (n * (n + 1) / 2); ` `         `  `        ``// sum of squares of 1st n  ` `        ``// natural numbers ` `        ``int` `sum_sq_n = (n * (n + 1) / 2) * ` `                        ``(2 * n + 1) / 3; ` `                     `  `        ``// required sum ` `        ``return` `(sum_n + sum_sq_n); ` `    ``} ` `     `  `    ``// Driver program to test above ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 5; ` `         `  `        ``Console.Write(``"Sum = "``+ ` `                    ``sumOfTheSeries(n)); ` `    ``} ` `} ` ` `  `// This code is contriubted by  ` `// vt_m `

## PHP

 ` `

Output:

```Sum = 70
```

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