The greatest coefficient in the expansion of (1+x)^(2n+1)

To find the greatest coefficient in the expansion of (1+x)^(2n+1), we need to understand the binomial theorem and the concept of binomial coefficients.

Binomial Theorem:
The binomial theorem states that for any positive integer n, the expansion of (a+b)^n can be written as the sum of the terms of the form C(n,k) * (a^k) * (b^(n-k)), where C(n,k) represents the binomial coefficient.

Binomial Coefficient:
The binomial coefficient C(n,k) represents the number of ways to choose k items from a set of n distinct items. It can be calculated using the formula C(n,k) = n! / (k! * (n-k)!), where n! denotes the factorial of n.

Expanding (1+x)^(2n+1):
In this case, we have (1+x)^(2n+1), which can be expanded using the binomial theorem. The expansion will consist of (2n+2) terms, as the power of (1+x) is (2n+1).

The general term in the expansion will be of the form C(2n+1,k) * (1^k) * (x^(2n+1-k)). Note that the coefficient of x in each term is 1^k, which is always 1. Therefore, to find the greatest coefficient, we need to find the term with the greatest binomial coefficient C(2n+1,k).

Finding the greatest coefficient:
To find the greatest coefficient, we need to determine the value of k that maximizes the binomial coefficient C(2n+1,k).

Key Points:
1. The binomial coefficient C(2n+1,k) is symmetric, meaning C(2n+1,k) = C(2n+1,2n+1-k). Therefore, we only need to consider values of k up to (2n+1)/2.
2. The binomial coefficient C(2n+1,k) is maximized when k is closest to (2n+1)/2. This can be intuitively understood by considering the symmetry of the coefficients.
3. Therefore, the greatest coefficient in the expansion of (1+x)^(2n+1) is C(2n+1,(2n+1)/2).

Conclusion:
In conclusion, the greatest coefficient in the expansion of (1+x)^(2n+1) is given by the binomial coefficient C(2n+1,(2n+1)/2). This coefficient can be calculated using the formula C(2n+1,k) = (2n+1)! / (((2n+1)/2)! * ((2n+1)-(2n+1)/2)!).