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Middle Term in the Binomial Expansion Series
How can you determine the middle term in the binomial expansion series? The middle term in a binomial expansion series is determined by whether the expansion has an odd or even number of terms.
The middle term is the (n+1)/2th term of the expansion if the number of terms is odd. For instance, the expansion of (a+b)^5 has 6 terms, making the middle term (5+1)/2 = 3rd term, or 10a^2b^3.
There are two intermediate terms—the n/2nd and (n/2)+1th terms of the expansion—if the number of terms is even. For instance, there are 5 terms in the expansion of (a+b)^4; the second term, which is 4ab^2, and the third term, which is 6a^2b^2, are the two middle terms.
We can use the following formula to determine the middle word in a binomial expansion:
T(k) = nCk * a ^ (n−k) * b^k
Where k is the index of the term, a and b are the coefficients of the terms being raised to the power, and n is the power to which the binomial is raised. When there are even numbers of terms, we substitute k = n/2 and k = (n/2)+1. When there is an odd number of terms we substitute k = (n+1)/2 to find the middle term, and we calculate the matching term to determine the middle term(s).
Approach
Now, let’s implement the theory we discussed. We will convert it into a step by step approach, which we will use to implement our code.
Specify the number of terms in the binomial expression as user input. You can also take this as user input.
Apart from the number of terms, Take the value of a and b also as user inputs.
Now based on the value of n (number of terms) entered by the user, we calculate the total number of terms to be n+1.
Calculate the index/indices of the middle terms, this will depend on whether the total number of terms is even or odd.
Initialize the value of the second middle term to 0, as we will need it only when the number of terms is even.
Run a loop for all the terms in the expansion, and calculate a value for each term. For calculating the value :
Calculate the binomial coefficient using the gamma function for each term. This calculation is based on the formula nCk= n!/(k!*(n−k)!)
Now once we have the binomial coefficient, calculate the value of the term by multiplying the coefficient with the appropriate power of a and b.
Now check if the index of the current term matches the index of the first middle term, if yes then print the value
Check again. If the total number of terms is even and the index of the current term matches the index of the second middle term, save the coefficient of that term and print its value.
C++ Code implementation to find the Middle term in the Binomial Expansion Series
Too much theory right? Now, let's do some code. Here is the C++ code implementation using the approach discussed above.
Example
#include <iostream>
#include <cmath>
using namespace std;
int main() {
int n = 4;
double a = 2, b = 3;
int numTerms = n + 1; // Total number of terms in the expansion
int middleIndex1, middleIndex2; // Indices of the middle term(s)
if (numTerms % 2 == 0) {
// Even number of terms, two middle terms
middleIndex1 = n/2;
middleIndex2 = (n/2) + 1;
}
else {
// Odd number of terms, one middle term
middleIndex1 = (n+1)/2;
middleIndex2 = 0; // Set the second middle-term index to 0
}
// Calculate the middle term(s)
double coeff1, coeff2;
for (int k = 1; k <= n; k++) {
double coeff = tgamma(n+1) / (tgamma(k+1) * tgamma(n-k+1)); // nCk coefficient
double term = coeff * pow(a, n-k) * pow(b, k);
if (k == middleIndex1) {
coeff1 = coeff;
cout << "The middle term is: " << term << endl;
}
if (k == middleIndex2) {
coeff2 = coeff;
cout << "The second middle term is: " << term << endl;
}
}
return 0;
}
Output
The middle term is: 216
Time Complexity: O(n)
Space Complexity: O(1)
Conclusion
In this article, we covered how to find the middle terms in the binomial expansion series given the total number of terms is odd or even. Hope you got a better clarity on the concept.
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