Explanation of the term from the end in the binomial expansion

In the given expression, we have (x^2 - 2/x^2)^(2n + 1). Let's break down the steps to find the term from the end in the binomial expansion of this expression.

Step 1: Expand the binomial
To expand the given binomial, we can use the binomial theorem. According to the binomial theorem, the expansion of (a + b)^n, where n is a positive integer, can be given by the formula:

(a + b)^n = C(n, 0) * a^n * b^0 + C(n, 1) * a^(n-1) * b^1 + C(n, 2) * a^(n-2) * b^2 + ... + C(n, n-1) * a^1 * b^(n-1) + C(n, n) * a^0 * b^n

Here, C(n, r) represents the binomial coefficient, given by the formula:

C(n, r) = n! / (r! * (n - r)!)

Step 2: Identify the term from the end
To find the term from the end in the binomial expansion, we need to determine the value of r. In this case, we are looking for the (2n + 1)th term from the end.

Step 3: Calculate the binomial coefficient
Using the formula for the binomial coefficient, we can calculate the value of C(n, r).

C(n, r) = (2n + 1)! / (r! * (2n + 1 - r)!)

Step 4: Substitute the values
Now, we substitute the calculated values of C(n, r), a, and b into the binomial expansion formula:

(x^2 - 2/x^2)^(2n + 1) = C(2n + 1, r) * (x^2)^(2n + 1 - r) * (-2/x^2)^r

Step 5: Simplify the expression
Simplifying the expression further, we have:

(x^2 - 2/x^2)^(2n + 1) = C(2n + 1, r) * x^(4n + 2 - 2r) * (-2)^r / x^(2r)

Step 6: Determine the term
To find the term from the end, we need to set the value of r such that the exponent of x is equal to -2. This can be represented as:

4n + 2 - 2r = -2

Solving this equation for r, we get:

r = (4n + 4) / 2

Simplifying further, we have:

r = 2n + 2

Therefore, the term from the end in the binomial expansion of (x^2 - 2/x^2)^(2n + 1) is given by:

C(2n + 1, 2n + 2) * x^(-2) * (-2)^(2n + 2)