(1%2B1%2Fx%2B1)(1%2B1%2Fx%2B2)(1%2B1%2Fx%2B3).jpeg "(1+1/x)(1+1/x+1)(1+1/x+2)(1+1/x+3)")
Exploring the Expansion of (1 + 1/x)(1 + 1/x + 1)(1 + 1/x + 2)(1 + 1/x + 3)
This article delves into the expansion of the expression (1 + 1/x)(1 + 1/x + 1)(1 + 1/x + 2)(1 + 1/x + 3), analyzing its pattern, and exploring its significance.
Step-by-Step Expansion
To understand the expression better, let's expand it step by step:
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Expanding the first two terms: (1 + 1/x)(1 + 1/x + 1) = 1 + 1/x + 1 + 1/x + 1/x² + 1/x = 1 + 3/x + 1/x² + 1
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Multiplying by the third term: (1 + 3/x + 1/x² + 1)(1 + 1/x + 2) = 1 + 3/x + 1/x² + 1 + 1/x + 3/x² + 2/x + 6/x² + 2/x³ + 2 = 4 + 6/x + 10/x² + 2/x³ + 2
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Finally, multiplying by the fourth term: (4 + 6/x + 10/x² + 2/x³ + 2)(1 + 1/x + 3) = 4 + 6/x + 10/x² + 2/x³ + 2 + 4/x + 6/x² + 10/x³ + 2/x⁴ + 12 + 18/x + 30/x² + 6/x³ + 6 = 22 + 26/x + 46/x² + 18/x³ + 2/x⁴ + 6
Therefore, the complete expanded form of the expression is:
(1 + 1/x)(1 + 1/x + 1)(1 + 1/x + 2)(1 + 1/x + 3) = 28 + 26/x + 46/x² + 18/x³ + 2/x⁴
Patterns and Observations
While the expansion might appear complex, we can observe some interesting patterns:
- Coefficients: The coefficients of the terms (28, 26, 46, 18, 2) don't follow a simple arithmetic progression.
- Exponents: The exponents of 'x' decrease in each term, starting from 0 and reaching -4.
- Symmetry: The expression exhibits a kind of symmetry in its expansion.
Significance
This type of expression often appears in various mathematical contexts, such as:
- Calculus: The expansion can be used to understand the behavior of functions involving rational expressions (expressions with variables in the denominator).
- Polynomial Analysis: The expression provides insights into the relationships between polynomials and their factors.
- Combinatorics: This type of expansion can be used to analyze combinatorial problems involving sequences.
Conclusion
The expansion of (1 + 1/x)(1 + 1/x + 1)(1 + 1/x + 2)(1 + 1/x + 3) reveals an interesting pattern and provides valuable insights into the behavior of rational expressions and polynomial manipulation. Understanding such expansions is crucial for tackling various mathematical problems in different fields.