Harmonic progression Sum

Last Updated : 10 Aug, 2022

Given the first element of the progression ‘a’, common difference between the element ‘d’ and number of terms in the progression ‘n’, where $n, d, a \in [1, 100000]$ . The task is to generate harmonic progression using the above set of information.
Examples:

Input : a = 12, d = 12, n = 5 
Output : Harmonic Progression :
         1/12 1/24 1/36 1/48 1/60 

         Sum of the generated harmonic 
         progression : 0.19
         
         Sum of the generated harmonic 
         progression using approximation :0.19

Arithmetic Progression : In an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
Harmonic Progression: A harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression.
Now, we need to generate this harmonic progression. We even have to calculate the sum of the generated sequence.
1. Generating of HP or 1/AP is a simple task. The Nth term in an AP = a + (n-1)d. Using this formula, we can easily generate the sequence.
2. Calculating the sum of this progression or sequence can be a time taking task. We can either iterate while generating this sequence or we could use some approximations and come up with a formula which would give us a value accurate up to some decimal places. Below is an approximate formula.

Sum = 1/d (ln(2a + (2n – 1)d) / (2a – d))

Please refer brilliant.org for details of above formula.
Below is implementation of above formula.

C++

// C++ code to generate Harmonic Progression
// and calculate the sum of the progression.
#include <bits/stdc++.h>
using namespace std;
 
// Function that generates the harmonic progression
// and calculates the sum of its elements by iterating.
double generateAP(int a, int d, int n, int AP[])
{
    double sum = 0;
    for (int i = 1; i <= n; i++)
    {
 
        // HP = 1/AP
        // In AP, ith term is calculated by a+(i-1)d;
        AP[i] = (a + (i - 1) * d);
 
        // Calculating the sum.
        sum += (double)1 / (double)((a + (i - 1) * d));
    }
    return sum;
}
 
// Function that uses riemann  sum method to calculate
// the approximate sum of HP in O(1) time complexity
double sumApproximation(int a, int d, int n)
{
    return log((2 * a + (2 * n - 1) * d) / 
                            (2 * a - d)) / d;
}
 
// Driver code
int main()
{
    int a = 12, d = 12, n = 5;
    int AP[n + 5] ;
     
    // Generating AP from the above data
    double sum = generateAP(a, d, n, AP);
 
    // Generating HP from the generated AP
    cout<<"Harmonic Progression :"<<endl;
    for (int i = 1; i <= n; i++)
        cout << "1/" << AP[i] << " ";
    cout << endl;
    string str = "";
    str = str + to_string(sum);
    str = str.substr(0, 4);
 
    cout<< "Sum of the generated"
            <<" harmonic progression : " << str << endl;
    sum = sumApproximation(a, d, n);
 
    str = "";
    str = str + to_string(sum);
    str = str.substr(0, 4);
    cout << "Sum of the generated "
    << "harmonic progression using approximation : "
                                        << str;
    return 0;
}
 
// This code is contributed by Rajput-Ji

Java

// Java code to generate Harmonic Progression
// and calculate the sum of the progression.
import java.util.*;
import java.lang.*;
 
class GeeksforGeeks {
 
    // Function that generates the harmonic progression
    // and calculates the sum of its elements by iterating.
    static double generateAP(int a, int d, int n, int AP[])
    {
        double sum = 0;
        for (int i = 1; i <= n; i++) {
 
            // HP = 1/AP
            // In AP, ith term is calculated by a+(i-1)d;
            AP[i] = (a + (i - 1) * d);
 
            // Calculating the sum.
            sum += (double)1 / (double)((a + (i - 1) * d));
        }
        return sum;
    }
 
    // Function that uses riemann  sum method to calculate
    // the approximate sum of HP in O(1) time complexity
    static double sumApproximation(int a, int d, int n)
    {
 
        return Math.log((2 * a + (2 * n - 1) * d) / 
                                 (2 * a - d)) / d;
    }
 
    public static void main(String args[])
    {
        int a = 12, d = 12, n = 5;
        int AP[] = new int[n + 5];
         
        // Generating AP from the above data
        double sum = generateAP(a, d, n, AP);
 
        // Generating HP from the generated AP
        System.out.println("Harmonic Progression :");
        for (int i = 1; i <= n; i++)
            System.out.print("1/" + AP[i] + " ");
        System.out.println();
        String str = "";
        str = str + sum;
        str = str.substring(0, 4);
 
        System.out.println("Sum of the generated" + 
                  " harmonic progression : " + str);
        sum = sumApproximation(a, d, n);
 
        str = "";
        str = str + sum;
        str = str.substring(0, 4);
        System.out.println("Sum of the generated " + 
           "harmonic progression using approximation : "
                                              + str);
    }
}

Python3

# Python3 code to generate Harmonic Progression
# and calculate the sum of the progression.
import math
 
# Function that generates the harmonic 
# progression and calculates the sum of
# its elements by iterating.
n = 5;
AP = [0] * (n + 5);
 
def generateAP(a, d, n):
    sum = 0;
    for i in range(1, n + 1):
         
        # HP = 1/AP
        # In AP, ith term is calculated
        # by a+(i-1)d;
        AP[i] = (a + (i - 1) * d);
 
        # Calculating the sum.
        sum += float(1) / float((a + (i - 1) * d));
    return sum;
 
# Function that uses riemann  sum method to calculate
# the approximate sum of HP in O(1) time complexity
def sumApproximation(a, d, n):
 
        return math.log((2 * a + (2 * n - 1) * d) /
                                 (2 * a - d)) / d;
 
# Driver Code
a = 12;
d = 12;
#n = 5;
 
# Generating AP from the above data
sum = generateAP(a, d, n);
 
# Generating HP from the generated AP
print("Harmonic Progression :");
for i in range(1, n + 1):
    print("1 /", AP[i], end = " ");
print("");
str1 = "";
str1 = str1 + str(sum);
str1 = str1[0:4];
 
print("Sum of the generated harmonic", 
               "progression :", str1);
sum = sumApproximation(a, d, n);
 
str1 = "";
str1 = str1 + str(sum);
str1 = str1[0:4];
print("Sum of the generated harmonic", 
      "progression using approximation :", str1);
 
# This code is contributed by mits     

C#

// C# code to generate Harmonic 
// Progression and calculate 
// the sum of the progression.
using System;
 
class GFG 
{
    // Function that generates 
    // the harmonic progression
    // and calculates the sum of
    // its elements by iterating.
    static double generateAP(int a, int d, 
                             int n, int []AP)
    {
        double sum = 0;
        for (int i = 1; i <= n; i++)
        {
 
            // HP = 1/AP
            // In AP, ith term is
            // calculated by a+(i-1)d;
            AP[i] = (a + (i - 1) * d);
 
            // Calculating the sum.
            sum += (double)1 / 
                   (double)((a + (i - 1) * d));
        }
        return sum;
    }
 
    // Function that uses riemann 
    // sum method to calculate
    // the approximate sum of HP
    // in O(1) time complexity
    static double sumApproximation(int a, 
                                   int d, int n)
    {
 
        return Math.Log((2 * a + 
                        (2 * n - 1) * d) / 
                        (2 * a - d)) / d;
    }
 
    // Driver code
    static void Main()
    {
        int a = 12, d = 12, n = 5;
        int []AP = new int[n + 5];
         
        // Generating AP from 
        // the above data
        double sum = generateAP(a, d, n, AP);
 
        // Generating HP from
        // the generated AP
        Console.WriteLine("Harmonic Progression :");
        for (int i = 1; i <= n; i++)
            Console.Write("1/" + AP[i] + " ");
        Console.WriteLine();
        String str = "";
        str = str + sum;
        str = str.Substring(0, 4);
 
        Console.WriteLine("Sum of the generated" + 
                " harmonic progression : " + str);
        sum = sumApproximation(a, d, n);
 
        str = "";
        str = str + sum;
        str = str.Substring(0, 4);
        Console.WriteLine("Sum of the generated " + 
        "harmonic progression using approximation : "
                                            + str);
    }
}
 
// This code is contributed by 
// ManishShaw(manishshaw1)

PHP

<?php
// PHP code to generate Harmonic Progression
// and calculate the sum of the progression.
 
    // Function that generates the harmonic progression
    // and calculates the sum of its elements by iterating.
    function generateAP($a, $d, $n, &$AP)
    {
        $sum = 0;
        for ($i = 1; $i <= $n; $i++) {
 
            // HP = 1/AP
            // In AP, ith term is calculated by a+(i-1)d;
            $AP[$i] = ($a + ($i - 1) * $d);
 
            // Calculating the sum.
            $sum += (double)1 / (double)(($a + ($i - 1) * $d));
        }
        return $sum;
    }
 
    // Function that uses riemann  sum method to calculate
    // the approximate sum of HP in O(1) time complexity
    function sumApproximation($a, $d, $n)
    {
 
        return log((2 * $a + (2 * $n - 1) * $d) / 
                                (2 * $a - $d)) / $d;
    }
 
// Drive main
        $a = 12;
        $d = 12;
        $n = 5;
        $AP = array_fill(0,$n + 5,0);
         
        // Generating AP from the above data
        $sum = generateAP($a, $d, $n, $AP);
 
        // Generating HP from the generated AP
        echo "Harmonic Progression :\n";
        for ($i = 1; $i <= $n; $i++)
            echo "1/".$AP[$i]." ";
        echo "\n";
        $str = "";
        $str = $str.strval($sum);
        $str = substr($str,0, 4);
 
        echo "Sum of the generated". 
                " harmonic progression : ".$str;
        $sum = sumApproximation($a, $d, $n);
 
        $str = "";
        $str = $str + strval($sum);
        $str = substr($str,0, 4);
        echo "\nSum of the generated ". 
        "harmonic progression using approximation : ".$str;
 
// this code is contributed by mits                                            
?>

Javascript

<script>
 
// JavaScript code to generate Harmonic Progression
// and calculate the sum of the progression.
 
    // Function that generates the harmonic progression
    // and calculates the sum of its elements by iterating.
    function generateAP(a, d, n, AP)
    {
        let sum = 0;
        for (let i = 1; i <= n; i++) {
   
            // HP = 1/AP
            // In AP, ith term is calculated by a+(i-1)d;
            AP[i] = (a + (i - 1) * d);
   
            // Calculating the sum.
            sum += 1 / ((a + (i - 1) * d));
        }
        return sum;
    }
   
    // Function that uses riemann  sum method to calculate
    // the approximate sum of HP in O(1) time complexity
    function sumApproximation(a, d, n)
    {
   
        return Math.log((2 * a + (2 * n - 1) * d) / 
                                 (2 * a - d)) / d;
    }
 
// Driver code    
          
        let a = 12, d = 12, n = 5;
        let AP = [];
           
        // Generating AP from the above data
        let sum = generateAP(a, d, n, AP);
   
        // Generating HP from the generated AP
        document.write("Harmonic Progression :" + "<br/>");
        for (let i = 1; i <= n; i++)
            document.write("1/" + AP[i] + " ");
        document.write("<br/>");
        let str = "";
        str = str + sum;
        str = str.substring(0, 4);
   
        document.write("Sum of the generated" + 
                  " harmonic progression : " + str + "<br/>");
        sum = sumApproximation(a, d, n);
   
        str = "";
        str = str + sum;
        str = str.substring(0, 4);
        document.write("Sum of the generated " + 
           "harmonic progression using approximation : "
                                              + str + "<br/>");
 
             
</script>

Output:

Harmonic Progression :
1/12 1/24 1/36 1/48 1/60 
Sum of the generated harmonic progression : 0.19
Sum of the generated harmonic progression using approximation :0.19

Time Complexity: O(n)

Auxiliary Space: O(n)

Harmonic Progression Program

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Harmonic progression Sum

C++

Java

Python3

C#

PHP

Javascript

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