(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^16)=(1-x^32)/(1-x)

Jasonbradley

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Jun 16, 2024

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A Beautiful Pattern: The Proof of (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^16)=(1-x^32)/(1-x)

This intriguing equation reveals a fascinating pattern in algebra. Let's explore its proof and understand the underlying concept.

The Power of Telescoping

The key to proving this equation lies in the telescoping method. This method involves manipulating terms in a way that most terms cancel out, leaving a simplified result. Here's how it works in this case:

1. Start with the left-hand side: (1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^16)

2. Multiply the first two terms: (1 + x + x^2 + x^3)(1+x^4)(1+x^8)(1+x^16)

3. Multiply the result by the next term (1+x^4): (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7)(1+x^8)(1+x^16)

4. Continue multiplying by the remaining terms, noticing the pattern: (1 + x + x^2 + x^3 + ... + x^15 + x^16 + x^17 + x^18 + x^19 + x^20 + x^21 + x^22 + x^23 + x^24 + x^25 + x^26 + x^27 + x^28 + x^29 + x^30 + x^31)(1+x^32)

5. Notice that when we multiply by (1+x^32), we essentially add each term with its corresponding power of x^32: 1 + x + x^2 + ... + x^31 + x^32 + x^33 + x^34 + ... + x^63 + x^64

6. This results in pairs of terms that cancel out: 1 + (x + x^32) + (x^2 + x^33) + ... + (x^31 + x^63) + x^64

7. The remaining terms simplify to: (1 - x^32)/(1 - x)

This is the right-hand side of our original equation, completing the proof.

The Essence of the Pattern

The pattern in this equation arises from the way we multiply the terms. Each multiplication introduces a new term with a power of x that is double the previous power. This leads to a chain of cancellations when we sum the terms, leaving only the first and last terms.

This equation is a great example of how simple algebraic manipulations can reveal beautiful and insightful patterns. It demonstrates the elegance of telescoping techniques and highlights the power of recognizing recurring patterns in mathematics.