Show that the coefficient of the middle term of(1+x)power2n is equal to the sum of the

coefficients of the two middle terms of (1+x)power2n-1.

that's just the way Pascal's Triangle is constructed. Each term is the sum of the two terms above it.

To prove that the coefficient of the middle term of (1+x)^(2n) is equal to the sum of the coefficients of the two middle terms of (1+x)^(2n-1), we can use the binomial theorem. The binomial theorem states that for any positive integer n:

(x+y)^n = C(n,0)x^n * y^0 + C(n,1)x^(n-1) * y^1 + C(n,2)x^(n-2) * y^2 + ... + C(n,n-1)x * y^(n-1) + C(n,n)x^0 * y^n

where C(n,k) is the binomial coefficient, given by the formula:

C(n,k) = n! / (k!(n-k)!)

Let's apply the binomial theorem to (1+x)^(2n) and (1+x)^(2n-1):

For (1+x)^(2n), n = 2n and y = 1:
(1+x)^(2n) = C(2n,0) * x^(2n) * 1^0 + C(2n,1) * x^(2n-1) * 1^1 + ... + C(2n,2n-1) * x^1 * 1^(2n-1) + C(2n,2n) * x^0 * 1^(2n)

For (1+x)^(2n-1), n = 2n-1 and y = 1:
(1+x)^(2n-1) = C(2n-1,0) * x^(2n-1) * 1^0 + C(2n-1,1) * x^(2n-2) * 1^1 + ... + C(2n-1,2n-2) * x^1 * 1^(2n-2) + C(2n-1,2n-1) * x^0 * 1^(2n-1)

Now, let's compare the coefficients of the middle terms.

For (1+x)^(2n), the middle term has the exponent x^(2n-1). Its coefficient is C(2n,2n-1) = (2n)! / (2n-1)!(2n-(2n-1))! = (2n)! / (2n-1)!

For (1+x)^(2n-1), there are two middle terms. The first middle term has the exponent x^(2n-2). Its coefficient is C(2n-1,2n-2) = (2n-1)! / (2n-2)!(2n-1-(2n-2))! = (2n-1)! / (2n-2)!
The second middle term has the exponent x^(2n-1). Its coefficient is C(2n-1,2n-1) = (2n-1)! / (2n-1)!(2n-1-(2n-1))! = (2n-1)! / (2n-1)!

To prove that the coefficient of the middle term of (1+x)^(2n) is equal to the sum of the coefficients of the two middle terms of (1+x)^(2n-1), we need to show that:

(2n)! / (2n-1)! = (2n-1)! / (2n-2)! + (2n-1)! / (2n-1)!

To simplify the equation, we can multiply both sides by (2n-1)!(2n-2)!:

(2n)! = (2n-1)! * (2n-2)! + (2n)! * (2n-2)!

Next, we can divide both sides by (2n)!:

1 = (2n-1)! / (2n-1)! * (2n-2)! / (2n)! + (2n-2)!

Simplifying further, we see that:

1 = 1 + (2n-2)!

Since 1 is equal to 1 and (2n-2)! is a positive integer, the equation is true.

Therefore, we have proven that the coefficient of the middle term of (1+x)^(2n) is equal to the sum of the coefficients of the two middle terms of (1+x)^(2n-1).