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 termfloat CalculateRatio(float m, float n){ // ratio will be tm/tn = (2*m - 1)/(2*n - 1) return (2 * m - 1) / (2 * n - 1);}// Driver codeint main(){ float m = 6, n = 2; cout << CalculateRatio(m, n); return 0;} |
Java
// Java code to calculate ratioimport java.io.*;class Nth { // function to calculate ratio of mth and nth termstatic 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 codeclass 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 ratiousing System;class Nth { // function to calculate ratio of mth and nth termfloat 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 termfunction 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> |
3.66667
Time Complexity: O(1)
Auxiliary Space: O(1)
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