Let x = 999, y = 1000 and z = 1001. Using the standard identity for x^3 + y^3 + z^3 - 3xyz, find the numerical value of x^3 + y^3 + z^3 - 3xyz.

Introduction / Context:
This algebra problem uses a well known identity for the sum of cubes of three numbers minus three times their product. When the three numbers form an arithmetic progression, especially of the form n - 1, n and n + 1, the expression simplifies beautifully. Recognising this pattern saves a lot of computation.

Given Data / Assumptions:
- x = 999.
- y = 1000.
- z = 1001.
- We must find x^3 + y^3 + z^3 - 3xyz.

Concept / Approach:
The key identity is x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). For numbers in arithmetic progression, such as n - 1, n and n + 1, this expression simplifies further. In fact, you can show that it equals 9n. Here, n is 1000, so we expect a multiple of 9000.

Step-by-Step Solution:
First compute x + y + z = 999 + 1000 + 1001 = 3000. Using the identity, x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx). Recognise that x, y and z are consecutive integers: x = n - 1, y = n, z = n + 1 with n = 1000. For this pattern, it can be shown that x^2 + y^2 + z^2 - xy - yz - zx = 3n. Thus, x^3 + y^3 + z^3 - 3xyz = (x + y + z)(3n) = 3000 * 3 = 9000.

Verification / Alternative check:
If you wish, you can confirm by direct computation using a calculator: 999^3 + 1000^3 + 1001^3 is a large number, but subtracting 3 * 999 * 1000 * 1001 yields 9000. This matches the identity based derivation and confirms the result.

Why Other Options Are Wrong:
Values like 1000, 1, 9 or 3000 do not match the identity. For example, 1000 is just one of the numbers, 1 and 9 are much too small compared with the magnitudes of the cubes, and 3000 ignores the additional factor of 3n in the identity. Only 9000 is consistent with a correct application of the formula.

Common Pitfalls:
A common error is to forget to subtract 3xyz and instead compute only x^3 + y^3 + z^3. Another pitfall is to misapply the identity or incorrectly assume that the expression always factors as (x + y + z)^3, which is not true. Careful use of the correct identity avoids these mistakes.

Final Answer:
The value of x^3 + y^3 + z^3 - 3xyz is 9000.