# 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 ])/2Given: 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 / non cross multiplying and solving, we get

d = 2 * aHence, 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; ` `} ` |

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## 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.. ` |

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## 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 ` |

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## 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. ` |

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## 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 ` `?> ` |

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**Output:**

3.66667

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