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

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++

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// C++ implementation to find Average
// of sum of first n natural even numbers
#include <bits/stdc++.h>
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;

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Java

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// 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'

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Python3

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# 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

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C#

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// 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'

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PHP

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<?php
// PHP implementation to find Average
// of sum of first n natural even numbers
  
// function to find average of 
// sum of first n even numbers
function avg_of_even_num($n)
{
    // sum of first n even numbers
    $sum = 0;
    for ($i = 1; $i <= $n; $i++) 
        $sum += 2 * $i;
  
    // calculating Average 
    return $sum / $n;
}
  
// Driver Code
$n = 9;
echo(avg_of_even_num($n));
  
// This code is contributed by Ajit.
?>

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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)

C++

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// CPP Program to find the average
// of sum of first n even numbers
#include <bits/stdc++.h>
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;
}

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Java

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// 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

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Python3

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# 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.

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C#

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// 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

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PHP

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<?php
// PHP Program to find the average
// of sum of first n even numbers
  
// Return the average of sum
// of first n even numbers
function avg_of_even_num($n)
{
    return $n + 1;
}
      
// Driver Code
$n = 8;
echo(avg_of_even_num($n));
  
// This code is contributed by Ajit.
?>

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Output:

9

Time Complexity : O(1)



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