Sum of first n odd numbers in O(1) Complexity

Given the sequence of odd numbers
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, ….
Find the sum of first n odd numbers

Examples:

Input : n = 2
Output : 4
Sum of first two odd numbers is 1 + 3 = 4.

Input : 5
Output : 25
Sum of first 5 odd numbers is 1 + 3 + 5 +
7 + 9 = 25

A simple solution is to iterate through all odd numbers.

C++

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// A naive CPP program to find sum of
// first n odd numbers
#include <iostream>
using namespace std;
  
// Returns the sum of first
// n odd numbers
int oddSum(int n)
{
    int sum = 0, curr = 1;
    for (int i = 0; i < n; i++) {
        sum += curr;
        curr += 2;
    }
    return sum;
}
  
// Driver function
int main()
{
    int n = 20;
    cout << " Sum of first " << n
         << " Odd Numbers is: " << oddSum(n);
    return 0;
}

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Java

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// Java program to find sum of
// first n odd numbers
import java.util.*;
  
class Odd
{   
    // Returns the sum of first
    // n odd numbers
    public static int oddSum(int n)
    {
        int sum = 0, curr = 1;
        for (int i = 0; i < n; i++) {
            sum += curr;
            curr += 2;
        }
        return sum;
    }
      
    // driver function
    public static void main(String[] args)
    {
        int n = 20;
        System.out.println(" Sum of first "+ n
        +" Odd Numbers is: "+oddSum(n));
    }
}
  
// This code is contributed by rishabh_jain

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Python3

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# Python3 program to find sum
# of first n odd numbers
  
def oddSum(n) :
    sum = 0
    curr = 1
    i = 0
    while i < n:
        sum = sum + curr
        curr = curr + 2
        i = i + 1
    return sum
  
# Driver Code
n = 20
print (" Sum of first" , n, "Odd Numbers is: ",
                                oddSum(n) ) 
  
# This code is contributed by rishabh_jain

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C#

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// C# program to find sum of
// first n odd numbers
using System;
  
class GFG {
      
    // Returns the sum of first
    // n odd numbers
    public static int oddSum(int n)
    {
        int sum = 0, curr = 1;
        for (int i = 0; i < n; i++) {
            sum += curr;
            curr += 2;
        }
          
        return sum;
    }
  
    // driver function
    public static void Main()
    {
        int n = 20;
        Console.WriteLine(" Sum of first " + n
            + " Odd Numbers is: " + oddSum(n));
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// A naive PHP program to find sum of
// first n odd numbers
  
// Returns the sum of first
// n odd numbers
function oddSum($n)
{
    $sum = 0; $curr = 1;
    for ($i = 0; $i < $n; $i++) 
    {
        $sum += $curr;
        $curr += 2;
    }
    return $sum;
}
  
// Driver Code
$n = 20;
echo " Sum of first ", $n
     , " Odd Numbers is: ", oddSum($n);
       
// This code is contributed by vt_m.
?>

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Output:

Sum of first 20 odd numbers is 400

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

 

An efficient solution is to use direct formula. To find the sum of first n odd numbers we can apply odd number theorem, it states that the sum of first n odd numbers is equal to the square of n.

&Sum;(2i – 1) = n2 where i varies from 1 to n

let n = 10, therefore sum of first 10 odd numbers is

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100

if we apply odd number theorem:

sum of first 10 odd numbers = n * n = 10 * 10 = 100.

Below is the implementation of the above approach:

C++

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// Efficient program to find sum of
// first n odd numbers
#include <iostream>
using namespace std;
  
// Returns the sum of first
// n odd numbers
int oddSum(int n)
{
    return (n * n);
}
  
// Driver function
int main()
{
    int n = 20;
    cout << " Sum of first " << n
         << " Odd Numbers is: " << oddSum(n);
    return 0;
}

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Java

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// Java program to find sum of
// first n odd numbers
import java.util.*;
  
class Odd
{   
    // Returns the sum of first
    // n odd numbers
    public static int oddSum(int n)
    {
        return (n * n);
    }
      
    // driver function
    public static void main(String[] args)
    {
        int n = 20;
        System.out.println(" Sum of first "+ n
        +" Odd Numbers is: "+oddSum(n));
    }
}
  
// This code is contributed by rishabh_jain

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Python3

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# Python3 program to find sum
# of first n odd numbers
  
def oddSum(n) :
    return (n * n);
  
# Driver Code
n = 20
print (" Sum of first" , n, "Odd Numbers is: ",
                               oddSum(n) ) 
  
# This code is contributed by rishabh_jain

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C#

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// C# program to find sum of
// first n odd numbers
using System;
  
class GFG {
      
    // Returns the sum of first
    // n odd numbers
    public static int oddSum(int n)
    {
        return (n * n);
    }
  
    // driver function
    public static void Main()
    {
        int n = 20;
        Console.WriteLine(" Sum of first " + n
            + " Odd Numbers is: " + oddSum(n));
    }
}
  
// This code is contributed by vt_m.

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PHP

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<?php
// Efficient program to find sum of
// first n odd numbers
  
// Returns the sum of first
// n odd numbers
function oddSum($n)
{
    return ($n * $n);
}
  
// Driver Code
$n = 20;
echo " Sum of first " , $n,
     " Odd Numbers is: ", oddSum($n);
  
// This code is contributed by vt_m.
?>

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Output:

Sum of first 20 odd numbers is 400

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

How does it work?
We can prove it using mathematical induction. We know it is true for n = 1 and n = 2 as sums are 1 and 4 (1 + 3) respectively.

Let it be true for n = k-1.

Sum of first k odd numbers = 
  Sum of first k-1 odd numbers + k'th odd number
= (k-1)*(k-1) + (2k - 1)
= k*k


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Improved By : vt_m