Dado el primer elemento de la progresión ‘a’, diferencia común entre el elemento ‘d’ y el número de términos de la progresión ‘n’, donde ![n, d, a \in [1, 100000]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-580899fb51f2d70a702c02cab19064c6_l3.png "Rendered by QuickLaTeX.com")
. La tarea es generar una progresión armónica utilizando el conjunto de información anterior.
Ejemplos:
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
Progresión aritmética: En una progresión aritmética (AP) o secuencia aritmética es una secuencia de números tal que la diferencia entre los términos consecutivos es constante.
Progresión armónica: Una progresión armónica (o secuencia armónica) es una progresión formada tomando los recíprocos de una progresión aritmética.
Ahora, necesitamos generar esta progresión armónica. Incluso tenemos que calcular la suma de la secuencia generada.
1. La generación de HP o 1/AP es una tarea sencilla. El N-ésimo término en un AP = a + (n-1)d. Usando esta fórmula, podemos generar fácilmente la secuencia.
2. Calcular la suma de esta progresión o secuencia puede llevar tiempo. Podemos iterar mientras generamos esta secuencia o podemos usar algunas aproximaciones y llegar a una fórmula que nos dé un valor preciso hasta algunos lugares decimales. A continuación se muestra una fórmula aproximada.
Suma = 1/d (ln(2a + (2n – 1)d) / (2a – d))
Consulte bright.org para obtener detalles de la fórmula anterior.
A continuación se muestra la implementación de la fórmula anterior.
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>
Producción:
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