# Average of first n even natural numbers

• Difficulty Level : Medium
• Last Updated : 12 Sep, 2022

Given a number n then Find the Average of first n even natural numbers
Ex.= 2 + 4 + 6 + 8 + 10 + 12 +………+ 2n.
Examples :

```Input  : 7
Output : 8
(2 + 4 + 6 + 8 + 10 + 12 + 14)/7 = 8

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

Naive Approach:- In this program iterate the loop , finding total sum of first n even numbers and divided by n.it take 0(N) time.

## C++

 `// C++ implementation to find Average``// of sum of first n natural even numbers``#include ``using` `namespace` `std;` `// function to find average of``// sum of first n even numbers``int` `avg_of_even_num(``int` `n)``{``    ``// sum of first n even numbers``    ``int` `sum = 0;``    ``for` `(``int` `i = 1; i <= n; i++)``        ``sum += 2*i;` `    ``// calculating Average``    ``return` `sum/n;``}` `// Driver Code``int` `main()``{``    ``int` `n = 9;``    ``cout << avg_of_even_num(n);``    ``return` `0;``}`

## Java

 `// java implementation to find Average``// of sum of first n natural even number``import` `java.io.*;` `class` `GFG {``    ` `    ``// function to find average of``    ``// sum of first n even numbers``    ``static` `int` `avg_of_even_num(``int` `n)``    ``{``    ` `    ``// sum of first n even numbers``    ``int` `sum = ``0``;``    ` `    ` `    ``for` `(``int` `i = ``1``; i <= n; i++)``        ``sum += ``2``*i;` `    ``// calculating Average``    ``return` `(sum / n);``    ``}``    ``public` `static` `void` `main (String[] args) {``    ` `    ``int` `n = ``9``;``    ``System.out.print(avg_of_even_num(n));``            ` `    ``}``}` `// this code is contributed by 'vt_m'`

## Python3

 `# Python3 implementation to``# find Average of sum of``# first n natural even``# number` `# Function to find average``# of sum of first n even``# numbers``def` `avg_of_even_num(n):``    ` `    ``# sum of first n even``    ``# numbers``    ``sum``=``0``    ``for` `i ``in` `range``(``1``, n ``+` `1``):``        ``sum``=``sum` `+` `2` `*` `i``    ` `    ``# calculating Average``    ``return` `sum` `/` `n` `n``=``9``print``(avg_of_even_num(n))` `# This code is contributed by upendra singh bartwal`

## C#

 `// C# implementation to find``// Average of sum of first``// n natural even number``using` `System;` `class` `GFG {``    ` `    ``// function to find average of``    ``// sum of first n even numbers``    ``static` `int` `avg_of_even_num(``int` `n)``    ``{``    ` `    ``// sum of first n even numbers``    ``int` `sum = 0;``    ` `    ``for` `(``int` `i = 1; i <= n; i++)``        ``sum += 2 * i;` `    ``// calculating Average``    ``return` `(sum / n);``    ``}``    ` `    ``// driver code``    ``public` `static` `void` `Main () {``    ` `    ``int` `n = 9;``    ``Console.Write(avg_of_even_num(n));``            ` `    ``}``}` `// This code is contributed by 'vt_m'`

## PHP

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## Javascript

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Output :

`  10`

Time Complexity : O(N)

Auxiliary Space: O(1) as it is using constant space

Method 2 :- The idea is the sum of first n even number is n(n+1), for find the Average of first n even numbers divide by n, hence formula is n(n + 1) / n = ( n + 1). i.e. Average of first n even numbers is n+1. it take 0(1) time.

`                  Avg of sum of N even natural number = (N + 1)`

Proof

```Sum of first n terms of an A.P.(Arithmetic Progression)
= (n/2) * [2*a + (n-1)*d].....(i)
where, a is the first term of the series and d is
the difference between the adjacent terms of the series.

Here, a = 2, d = 2, applying these values to eq.(i), get
Sum = (n/2) * [2*2 + (n-1)*2]
= (n/2) * [4 + 2*n - 2]
= (n/2) * (2*n + 2)
= n * (n + 1)

finding the Avg so divided by n = n*(n+1)/n
= (n+1)```

## C++

 `// CPP Program to find the average``// of sum of first n even numbers``#include ``using` `namespace` `std;` `// Return the average of sum``// of first n even numbers``int` `avg_of_even_num(``int` `n)``{``    ``return` `n+1;``}``    ` `// Driver Code``int` `main()``{``    ``int` `n = 8;``    ``cout << avg_of_even_num(n) << endl;``    ``return` `0;``}`

## Java

 `// Java Program to find the average``// of sum of first n even numbers``import` `java.io.*;` `class` `GFG``{` `    ``// Return the average of sum``    ``// of first n even numbers``    ``static` `int` `avg_of_even_num(``int` `n)``    ``{``        ``return` `n + ``1``;``    ``}``    ` `    ``public` `static` `void` `main (String[] args) {``        ` `        ``int` `n = ``8``;``        ``System.out.println(avg_of_even_num(n));``        ` `    ``}``}` `// This code is contributed by vt_m`

## Python3

 `# Python 3 Program to``# find the average``# of sum of first n``# even numbers` `# Return the average of sum``# of first n even numbers``def` `avg_of_even_num(n) :``    ` `    ``return` `n``+``1``    ` `     ` `# Driven Program``n ``=` `8``print``(avg_of_even_num(n))`  `# This code is contributed``# by Nikita Tiwari.`

## C#

 `// C# Program to find the average``// of sum of first n even numbers``using` `System;` `class` `GFG {` `    ``// Return the average of sum``    ``// of first n even numbers``    ``static` `int` `avg_of_even_num(``int` `n)``    ``{``        ``return` `n + 1;``    ``}``    ` `    ``// driver code   ``    ``public` `static` `void` `Main () {``        ` `        ``int` `n = 8;``        ``Console.Write(avg_of_even_num(n));``        ` `    ``}``}` `// This code is contributed by vt_m`

## PHP

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## Javascript

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Output:

`9`

Time Complexity: O(1)

Auxiliary Space: O(1)

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