# Average of first n even natural numbers

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
```

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

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++ 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 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 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# 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' `

 ` `

Output :

```  10
```

Time Complexity : O(N)

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)
```
 `// 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 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 `

 `# 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# 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 `

 ` `

Output:

```9
```

Time Complexity : O(1)

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Improved By : jit_t

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