# 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
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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++ 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 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 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# 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. `

 ` `

Output:
```3.66667
```

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Improved By : inderDuMCA, vt_m, Shivi_Aggarwal

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