Sum of the series 1 + (1+3) + (1+3+5) + (1+3+5+7) + ...... + (1+3+5+7+...+(2n-1))

20 Apr, 2021
Given a positive integer n. The problem is to find the sum of the given series 1 + (1+2) + (1+2+3) + (1+2+3+4) + ...... + (1+2+3+4+...+n), where i-th term in the series is the sum of first i odd natural numbers.Examples:  
Input : n = 2
Output : 5
(1) + (1+3) = 5

Input : n = 5
Output : 55
(1) + (1+3) + (1+3+5) + (1+3+5+7) + (1+3+5+7+9) = 55
 
Naive Approach: Using two loops get the sum of each i-th term and then add those sum to the final sum. 
C++
Java
Python3
C#
php
Javascript
// C++ implementation to find the// sum of the given series#include <bits/stdc++.h> using namespace std; // functionn to find the// sum of the given seriesint sumOfTheSeries(int n){    int sum = 0;    for (int i = 1; i <= n; i++) {         // first term of each i-th term        int k = 1;        for (int j = 1; j <= i; j++) {            sum += k;             // next term            k += 2;        }    }     // required sum    return sum;} // Driver programint main(){    int n = 5;    cout << "Sum = "         << sumOfTheSeries(n);    return 0;}
// Java implementation to find// the sum of the given seriesimport java.util.*; class GFG {         // functionn 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 = 1;            for (int j = 1; j <= i; j++)            {                sum += k;                      // next term                k += 2;            }        }              // required sum        return sum;    }     /* Driver program */    public static void main(String[] args)    {         int n = 5;         System.out.println("Sum = " +                        sumOfTheSeries(n));    }} // This code is contributed by Arnav Kr. Mandal.
# Python3 implementation to find# the sum of the given series # functionn to find the sum# of the given seriesdef sumOfTheSeries( n ):    sum = 0    for i in range(1, n + 1):                 # first term of each i-th term        k = 1        for j in range(1,i+1):            sum += k                         # next term            k += 2                 # required sum    return sum     # Driver programn = 5print("Sum =", sumOfTheSeries(n)) # This code is contributed by "Sharad_Bhardwaj".
// C# implementation to find// the sum of the given seriesusing System; class GFG {     // functionn 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 = 1;            for (int j = 1; j <= i; j++) {                sum += k;                 // next term                k += 2;            }        }         // required sum        return sum;    }     /* Driver program */    public static void Main()    {        int n = 5;        Console.Write("Sum = " +                     sumOfTheSeries(n));    }} // This code is contributed by vt_m.
<?php // php implementation to find the// sum of the given series // functionn to find the// sum of the given seriesfunction sumOfTheSeries($n){    $sum = 0;    for ($i = 1; $i <= $n; $i++) {          // first term of each i-th term        $k = 1;        for ($j = 1; $j <= $i; $j++) {            $sum += $k;              // next term            $k += 2;        }    }      // required sum    return $sum;}  // Driver program    $n = 5;    echo "Sum = "         . sumOfTheSeries($n); // This code is contributed by Sam007?>
<script> // Javascript implementation to find the// sum of the given series // functionn to find the// sum of the given seriesfunction sumOfTheSeries(n){    let sum = 0;    for (let i = 1; i <= n; i++) {          // first term of each i-th term        let k = 1;        for (let j = 1; j <= i; j++) {            sum += k;              // next term            k += 2;        }    }      // required sum    return sum;}  // Driver program    let n = 5;    document.write("Sum = " + sumOfTheSeries(n)); // This code is contributed by gfgking </script>
Output:  
Sum = 55
Efficient Approach: Let an be the n-th term of the given series.  
an = (1 + 3 + 5 + 7 + (2n-1))
   = sum of first n odd numbers
   = n2
Refer this post for the proof of above formula.Now,  
Refer this post for the proof of above formula. 
C++
Java
Python3
C#
PHP
Javascript
// C++ implementation to find the sum// of the given series#include <bits/stdc++.h>using namespace std; // functionn to find the sum// of the given seriesint sumOfTheSeries(int n){    // required sum    return (n * (n + 1) / 2) *            (2 * n + 1) / 3;} // Driver program to test aboveint main(){    int n = 5;    cout << "Sum = "         << sumOfTheSeries(n);    return 0;}
// Java implementation to find// the sum of the given seriesimport java.io.*; class GfG {     // function to find the sum// of the given seriesstatic int sumOfTheSeries(int n){    // required sum    return (n * (n + 1) / 2) *            (2 * n + 1) / 3;}      // Driver program to test abovepublic static void main (String[] args){    int n = 5;         System.out.println("Sum = "+                sumOfTheSeries(n)); } } // This code is contributed by Gitanjali.
# Python3 implementation to find# the sum of the given series # functionn to find the sum# of the given seriesdef sumOfTheSeries( n ):         # required sum    return int((n * (n + 1) / 2) *            (2 * n + 1) / 3)             # Driver program to test aboven = 5print("Sum =", sumOfTheSeries(n)) # This code is contributed by "Sharad_Bhardwaj".
// C# implementation to find// the sum of the given seriesusing System; class GfG {     // function to find the sum    // of the given series    static int sumOfTheSeries(int n)    {        // required sum        return (n * (n + 1) / 2) *                      (2 * n + 1) / 3;    }     // Driver program to test above    public static void Main()    {        int n = 5;         Console.Write("Sum = " +                   sumOfTheSeries(n));    }} // This code is contributed by vt_m.
<?php// PHP implementation to find the sum// of the given series // functionn to find the sum// of the given seriesfunction sumOfTheSeries($n){         // required sum    return ($n * ($n + 1) / 2) *              (2 * $n + 1) / 3;}     // Driver Code    $n = 5;    echo "Sum = "        . sumOfTheSeries($n);         // This code is contributed by Sam007?>
<script> // JavaScript program to find// the sum of the given series // function to find the sum    // of the given series    function sumOfTheSeries(n)    {        // required sum        return (n * (n + 1) / 2) *                      (2 * n + 1) / 3;    } // Driver Code         let n = 5;           document.write("Sum = " +                   sumOfTheSeries(n));                 // This code is contributed by avijitmondal1998.</script>
Output:  
Sum = 55
 
Sam007
avijitmondal1998
gfgking
number-theory
series
Mathematical
number-theory
Mathematical
series
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