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Exploring the Mathematical Expression 1/n! - 1/(n-1)! and Its Simplification
Exploring the Mathematical Expression 1/n! - 1/(n-1)! and Its Simplification
The expression 1/(n!)-1/(n-1)!1-n-n2/(n!) is a common algebraic expression often encountered in various mathematical calculations, particularly in problems related to probability, combinatorics, and series expansions. This article explores the simplification and interpretation of this expression, providing a deeper understanding and practical examples.
Understanding the Expression
The expression 1/(n!)-1/(n-1)!1-n-n2/(n!) involves the factorial of n, denoted as n!, which is the product of all positive integers less than or equal to n. For example, 5! 5 x 4 x 3 x 2 x 1 120.
Simplification Steps
Let's break down the expression into steps for better clarity:
First, express the given expression in a more detailed form: 1(n!)-1(n-1)!-1(n!' )#x21D2;1-n!-n!n! Simplify the numerator: 1(n!)-1(n-1)!-1(n#x2212;1!)#x21D2;1-n-n2n! The simplified form is thus expressed as: 1(n!)-1/(n-1)!1-n-n2(n!)Example Calculation
Let's pick a numerical example to demonstrate the application of the expression.
Pick a Couple Numbers and Run an Example
Take n 4 as an example:
1(4!)-1/(3)!1(24)-1/(6) Calculate each term separately: 124-1/6 Convert the terms to a common denominator: 124-424124-0.1667 Subtract the terms: 124-424124-0.1667#x21D2;-0.0417 Generalize as: 1(n1!)-1/(n-1)1!1-n1#xD7;n(n1!)For further complexity, let's simplify the final expression in a more abstract form:
1(n1!)-1/(n-1)1!1-n1#xD7;n(n1!)Conclusion
The expression 1/(n!)-1/(n-1)!1-n-n2n! significantly simplifies to 1-n1#xD7;n(n!). This form is not only more manageable but also more direct, providing us with a clear pathway to solving similar problems and ensuring accurate calculations in various mathematical contexts.