If x + 1 / x = 2, find the value of x^21 + 1 / x^21.

Introduction / Context:

This problem looks complex because it asks for x^21 + 1 / x^21, but the given condition x + 1 / x = 2 is very special and makes the evaluation easy. Recognising when a condition forces x to take a simple value is an important skill in algebra and aptitude questions. In this case, the expression simplifies dramatically once you solve for x.

Given Data / Assumptions:

• x + 1 / x = 2.
• x is nonzero so that 1 / x is defined.
• You must find x^21 + 1 / x^21.

Concept / Approach:

Start by solving the equation x + 1 / x = 2 for x. Multiplying both sides by x gives a quadratic equation. In this particular case, the quadratic has a repeated root, which makes x extremely simple. Once you know x, raising x to any positive integer power and adding its reciprocal becomes trivial.

Step-by-Step Solution:

Verification / Alternative check:

Because the equation x + 1 / x = 2 has a unique solution x = 1, there is no ambiguity. You can check the original condition by substituting x = 1: 1 + 1 / 1 = 1 + 1 = 2, which matches. Therefore, any expression in terms of x can be evaluated simply by substituting x = 1, and no further checks are necessary.

Why Other Options Are Wrong:

The values 0, 1, 3, and 21 might appear plausible if one incorrectly applies identities for powers of x + 1 / x or miscalculates higher powers. However, once x is known to be 1, all positive integer powers of x are 1, and the sum with its reciprocal is exactly 2. Any other answer contradicts this simple observation.

Common Pitfalls:

Some learners try to use recurrence relations or power formulas for x + 1 / x without noticing that the quadratic equation has the trivial root x = 1. This leads to unnecessary work and can cause errors. Always check first whether the given condition directly fixes x to a simple number like 1 or −1.

Final Answer:

The value of x^21 + 1 / x^21 is 2.