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Magazine Chapter 8: Problem 1
Find the generating function for the finite sequence \(2,2,\) \(2,2,2,2 .\)
Short Answer
Expert verified
The generating function is \ G(x) = 2 \frac{x^6 - 1}{x - 1}. \
Step by step solution
01
Define the given sequence
Consider the given finite sequence: 2, 2, 2, 2, 2, 2.
02
Write the general form of a generating function
A generating function for a sequence \( a_0, a_1, a_2, \ldots \, a_n \) is given by: \[ G(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n. \]
03
Substitute the sequence values into the generating function form
For the given sequence (2, 2, 2, 2, 2, 2), the generating function is: \[ G(x) = 2 + 2x + 2x^2 + 2x^3 + 2x^4 + 2x^5. \]
04
Factor the common factor out
Factor out the 2 from each term: \[ G(x) = 2(1 + x + x^2 + x^3 + x^4 + x^5). \]
05
Recognize the geometric series
Identify that the expression inside the parentheses is a finite geometric series with the first term 1 and common ratio x: \[ 1 + x + x^2 + x^3 + x^4 + x^5. \]
06
Apply the geometric series formula
Use the formula for the sum of the first n+1 terms of a geometric series: \[ \sum_{k=0}^{n} ar^k = a \frac{r^{n+1} - 1}{r-1}. \] Here, \( a = 1, \) \( r = x, \) and \( n = 5. \) Therefore, \[ 1 + x + x^2 + x^3 + x^4 + x^5 = \frac{x^6 - 1}{x - 1}. \]
07
Substitute the simplified series back into the generating function
Substitute the simplified sum back into the generating function expression: \[ G(x) = 2 \frac{x^6 - 1}{x - 1}. \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
finite sequence
A finite sequence is simply a list of numbers that has a specific number of terms. In our given problem, the sequence is 2, 2, 2, 2, 2, 2 which clearly has six terms.
Finite sequences can be contrasted with infinite sequences, which go on forever without terminating.
Finite sequences are useful because they allow us to use specific mathematical tools and techniques to analyze their properties.
For example, the generating function we derive helps us find patterns and manipulations within the given finite sequence.
In many real-world problems, like finance or scientific measurements, finite sequences are abundant and manageable.
Finite sequences can be contrasted with infinite sequences, which go on forever without terminating.
Finite sequences are useful because they allow us to use specific mathematical tools and techniques to analyze their properties.
For example, the generating function we derive helps us find patterns and manipulations within the given finite sequence.
In many real-world problems, like finance or scientific measurements, finite sequences are abundant and manageable.
geometric series
A geometric series is a series of terms that each term after the first is found by multiplying the previous one by a constant ratio. This constant ratio is called the common ratio.
In our problem, we identified that the sequence 1 + x + x^2 + x^3 + x^4 + x^5 is a geometric series with a common ratio of x.
Geometric series have practical applications in various fields.
For example, in finance, geometric series are used in calculating compound interest.
Using the formula for the sum of the first n+1 terms of a geometric series: \[ \sum_{k=0}^{n} ar^k = a \frac{r^{n+1} - 1}{r-1} \] we manage to express our sum in a simplified form.
This is a powerful technique because it lets us convert a long sum into a simple fraction.
In our problem, we identified that the sequence 1 + x + x^2 + x^3 + x^4 + x^5 is a geometric series with a common ratio of x.
Geometric series have practical applications in various fields.
For example, in finance, geometric series are used in calculating compound interest.
Using the formula for the sum of the first n+1 terms of a geometric series: \[ \sum_{k=0}^{n} ar^k = a \frac{r^{n+1} - 1}{r-1} \] we manage to express our sum in a simplified form.
This is a powerful technique because it lets us convert a long sum into a simple fraction.
discrete mathematics
Discrete mathematics studies mathematical structures that are fundamentally discrete rather than continuous.
This means we deal with countable, distinct elements.
Our exercise centers around a finite sequence and demonstrates the manipulation and representation of discrete elements using generating functions.
Generating functions are a critical tool in discrete mathematics.
They convert sequences into a form that can be manipulated algebraically.
Specifically, for a finite sequence, the generating function provides a compact expression that captures all the information about the sequence.
Discrete mathematics is foundational in a variety of fields, including computer science, cryptography, and combinatorics.
Through exercises like these, we develop a deeper understanding of finite structures and their properties.
This means we deal with countable, distinct elements.
Our exercise centers around a finite sequence and demonstrates the manipulation and representation of discrete elements using generating functions.
Generating functions are a critical tool in discrete mathematics.
They convert sequences into a form that can be manipulated algebraically.
Specifically, for a finite sequence, the generating function provides a compact expression that captures all the information about the sequence.
Discrete mathematics is foundational in a variety of fields, including computer science, cryptography, and combinatorics.
Through exercises like these, we develop a deeper understanding of finite structures and their properties.
