# 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++

`// 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;` `}` |

## 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

`<?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` `?>` |

## Javascript

`<script>` `// JavaScript 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` ` ` `let m = 6, n = 2;` ` ` `document.write(CalculateRatio(m, n));` `// This code is contributed by Surbhi Tyagi.` `</script>` |

**Output:**

3.66667