Exploring the Mathematical Pattern of x and 1/x

Mathematics often reveals fascinating patterns and relationships through algebraic identities and exponential functions. This article will delve into the pattern observed when a given expression involves the form of x and 1/x. Specifically, we will explore how to simplify expressions like x - 1/x p and further extend the analysis to higher powers.

Given Expression: x - 1/x p

We are given the equation x - 1/x p. Our goal is to find a simplified form for x - 1/x^{n} for any nonnegative integer n.

Substitution and Identity Application

Let's begin by substituting x^2 a. Using the identity for the sum of cubes, a^3 b^3 (a b)(a^2 - ab b^2), we can simplify the given expression.

Step 1: Substitution and Identity

First, substitute x^2 a into the expression:

[x - 1/x a^{3/2} - (1/x)^{3/2}]

Using the identity for the sum of cubes:

[a^3 - b^3 (a - b)(a^2 ab b^2)]

Substitute a x^2 and b 1/x^2:

[x - 1/x (a - 1)(a^2 a cdot 1/a 1/a^2)]

[x - 1/x (a - 1)(a^2 1/a^2 1/a)]

Replace (a) back with (x^2):

[x - 1/x (x^2 - 1)((x^2)^2 1/(x^2)^2 1/x^2)]

[x - 1/x (x^2 - 1)(x^4 1/x^4 1/x^2)]

Generalizing to Higher Powers

To generalize the expression for any nonnegative integer n, we can use a recursive sequence. Define (a_n x^n - 1/x^n). We will prove that (a_{n 2} - p cdot a_{n 1} a_n 0).

Proof by Recursion

Let's prove this by induction:

Basis Step (n 0 and n 1)

[a_0 x^0 - 1/x^0 1 - 1 0]

[a_1 x^1 - 1/x^1 x - 1/x p]

Inductive Step

Assume the statement is true for n k and n k 1. We need to show it holds for n k 2:

[a_{k 2} - p cdot a_{k 1} a_k 0]

Using the recursive relation:

[a_{k 2} p cdot a_{k 1} - a_k]

[a_{k 2} - p cdot a_{k 1} a_k p cdot a_{k 1} - a_k - p cdot a_{k 1} a_k 0]

This completes the inductive step, thus proving the statement for all nonnegative integers n.

Example Computations

Let's compute the first few terms:

[a_0 0]

[a_1 p]

[a_2 p^2 - 2]

[a_3 p^3 - 3p]

[a_4 p^4 - 4p^2]

[a_5 p^5 - 5p^3]

[a_6 p^6 - 6p^4]

Conclusion

The pattern observed in the sequence of expressions for (x) and (1/x) can be generalized using recursive relationships. Understanding these patterns can help in simplifying and solving complex expressions in algebra. Enjoy delving into the world of mathematics!