C++ Given the ratio of the sum of an arithmetic sequence, calculate the ratio of the Mth term and the Nth term

Discuss a problem in which the ratio of the sum of m and n terms of A.P is given. We need to find the ratio of the mth term to the nth term.

Input: m = 8, n = 4
Output: 2.142

Input: m = 3, n = 2
Output: 1.666

Input: m = 7, n = 3
Output: 2.6

Solution method

To find the ratio of the mth term to the nth term using code, we need to simplify the formula. Let S_m be the sum of the first m terms and S_n be the sum of the first n terms of A.P.

a - the first item,

d - the tolerance,

is given, S_m / S_n = m^{2 / n2}

#S ​​formula, S_m = (m/2)[ 2*a (m -1)*d]

m² / n² = (m/2)[ 2*a (m-1)*d ] / (n/2)[ 2*a (n-1)*d ]

m / n = [ 2*a (m-1) *d ] / [ 2*a (m-1) * d ]

Use cross multiplication,

n[ 2*a (m−1)*d ] = m[ 2*a (n− 1)*d]

2an mnd - nd = 2am mnd - md

2an - 2am = nd - md

(n - m)2a = (n-m)d

2an - 2am = nd 2a

The formula for the m

^{item is:}

        T

m _{= a (m-1)d}

The ratio of the m

^{th item to the nth item is,}

          T

m _{/ T}n _{= a (m -1)d / a (n-1)d}

Replace d with 2a,

            Tm / Tn = a (m-1)*2a / a (n-1)*2a

tm / tn = a (1 2m − 2) / a (1 2N − 2)

#TM / TN = 2m -1 / 2n -1

So now we have a simple formula for finding the ratio of the m

th

th term to the n^thth term. Let's take a look at the C code. Example

C code for the above method

#include <bits/stdc++.h>
using namespace std;
int main(){
    float m = 8, n = 4;
    // calculating ratio by applying formula.
    float result = (2 * m - 1) / (2 * n - 1);
    cout << "The Ratio of mth and nth term is: " << result;
    return 0;
}

Output

The ratio of mth and nth term is: 2.14286

Conclusion

In this tutorial , we discussed a problem of finding the ratio of the m-th term to the n-th term under a given sum ratio. We solved this problem by simplifying the formula for the sum of m terms and the formula for the m-th semester. We also discussed a C program to solve this problem and we can implement it using programming languages ​​like C, Java, Python etc. We hope you found this tutorial helpful.

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