# 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; ` `} ` |

*chevron_right*

*filter_none*

## 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.. ` |

*chevron_right*

*filter_none*

## 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 ` |

*chevron_right*

*filter_none*

## 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. ` |

*chevron_right*

*filter_none*

## 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 ` `?> ` |

*chevron_right*

*filter_none*

**Output:**

3.66667

## Recommended Posts:

- Find the number which when added to the given ratio a : b, the ratio changes to c : d
- Deriving the expression of Fibonacci Numbers in terms of golden ratio
- Sum of two numbers if the original ratio and new ratio obtained by adding a given number to each number is given
- Ratio of mth and nth term in an Arithmetic Progression (AP)
- Find nth Fibonacci number using Golden ratio
- Program to calculate the profit sharing ratio
- Find if it is possible to get a ratio from given ranges of costs and quantities
- Program to find the common ratio of three numbers
- Program to find the count of coins of each type from the given ratio
- Divide an isosceles triangle in two parts with ratio of areas as n:m
- Section formula (Point that divides a line in given ratio)
- Find amount to be added to achieve target ratio in a given mixture
- Count the number of rectangles such that ratio of sides lies in the range [a,b]
- Find the ratio of number of elements in two Arrays from their individual and combined average
- Ratio of the distance between the centers of the circles and the point of intersection of two direct common tangents to the circles

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.