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

• Last Updated : 07 Jul, 2022

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++

 `// C++ code to calculate ratio``#include ``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;``}`

## Java

 `// 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..`

## Python3

 `# 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`

## C#

 `// 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.`

## PHP

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## Javascript

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Output:

`3.66667`

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

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