Ratio of mth and nth terms of an A. P. with given ratio of sums

Given that the ratio to sum of first m and n terms of an A.P. with first term ‘a’ and commond difference ‘d’ is m^2:n^2. The task is to find the ratio of mth and nth term of this A.P.

Examples:

Input: m = 3, n = 2
Output: 1.6667

Input: m = 5, n = 3
Output: 1.8


Approach:

Let the Sum of first m and n terms be denoted by Sm and Sn respectively.
Also, let the mth and nth term be denoted by tm and tn respectively.

Sm = (m * [ 2*a + (m-1)*d ])/2
Sn = (n * [ 2*a + (n-1)*d ])/2

Given: Sm / Sn = m^2 / n^2
Hence, ((m * [ 2*a + (m-1)*d ])/2) / ((n * [ 2*a + (n-1)*d ])/2) = m^2 / n^2
=> (2*a + (m-1)*d) / (2*a + (n-1)*d) = m / n

on cross multiplying and solving, we get
d = 2 * a

Hence, the mth and nth terms can be written as:

mth term = tm = a +(m-1)*d = a + (m-1)*(2*a)
nth term = tn = a +(n-1)*d = a + (n-1)*(2*a)

Hence the ratio will be:
tm / tn = (a + (m-1)*(2*a)) / (a + (n-1)*(2*a))
tm / tn = (2*m – 1) / (2*n – 1)

Below is the required implementation:

C++

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// C++ code to calculate ratio
#include <bits/stdc++.h>
using namespace std;
  
// function to calculate ratio of mth and nth term
float CalculateRatio(float m, float n)
{
    // ratio will be tm/tn = (2*m - 1)/(2*n - 1)
    return (2 * m - 1) / (2 * n - 1);
}
  
// Driver code
int main()
{
    float m = 6, n = 2;
    cout << CalculateRatio(m, n);
  
    return 0;
}

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Java

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// Java code to calculate ratio
import java.io.*;
  
class Nth {
      
// function to calculate ratio of mth and nth term
static float CalculateRatio(float m, float n)
{
    // ratio will be tm/tn = (2*m - 1)/(2*n - 1)
    return (2 * m - 1) / (2 * n - 1);
}
}
  
// Driver code
class GFG {
      
    public static void main (String[] args) {
    float m = 6, n = 2;
    Nth a=new Nth();
System.out.println(a.CalculateRatio(m, n));
  
    }
}
  
// this code is contributed by inder_verma..

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Python3

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# Python3 program to calculate ratio 
  
# function to calculate ratio 
# of mth and nth term 
def CalculateRatio(m, n): 
  
    # ratio will be tm/tn = (2*m - 1)/(2*n - 1) 
    return (2 * m - 1) / (2 * n - 1); 
  
# Driver code 
if __name__=='__main__':
    m = 6;
    n = 2
    print (float(CalculateRatio(m, n))); 
  
# This code is contributed by 
# Shivi_Aggarwal

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C#

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// C# code to calculate ratio
using System;
  
class Nth {
      
// function to calculate ratio of mth and nth term
float CalculateRatio(float m, float n)
{
    // ratio will be tm/tn = (2*m - 1)/(2*n - 1)
    return (2 * m - 1) / (2 * n - 1);
}
  
    // Driver code
    public static void Main () {
    float m = 6, n = 2;
    Nth a=new Nth();
Console.WriteLine(a.CalculateRatio(m, n));
  
    }
}
// this code is contributed by anuj_67.

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PHP

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<?php
// PHP code to calculate ratio
  
// function to calculate ratio 
// of mth and nth term
function CalculateRatio( $m, $n)
{
    // ratio will be tm/tn = (2*m - 1)/(2*n - 1)
    return (2 * $m - 1) / (2 * $n - 1);
}
  
// Driver code
  
$m = 6; $n = 2;
echo CalculateRatio($m, $n);
  
// This code is contributed 
// by inder_verma
?>

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Output:

3.66667


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