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In Exercises \(3-8,\) by a closed form we mean an algebraic expression not involving a summation over a range of values or the use of ellipses. $$\begin{array}{l}{\text { Find a closed form for the generating function for each of }} \\ {\text { these sequences. (For each sequence, use the most obvi- }} \\ {\text { ous choice of a sequence that follows the pattern of the }} \\ {\text { initial terms listed.) }}\end{array}$$ $$ \begin{array}{l}{\text { a) } 0,2,2,2,2,2,2,0,0,0,0,0, \ldots} \\ {\text { b) } 0,0,0,1,1,1,1,1,1, \ldots} \\ {\text { c) } 0,1,0,0,1,0,0,1,0,0,1, \ldots} \\\ {\text { d) } 2,4,8,16,32,64,128,256, \ldots}\end{array} $$ $$ \begin{array}{l}{\text { e) }\left(\begin{array}{l}{7} \\\ {0}\end{array}\right),\left(\begin{array}{l}{7} \\\ {1}\end{array}\right),\left(\begin{array}{l}{7} \\ {2}\end{array}\right), \ldots,\left(\begin{array}{l}{7} \\ {7}\end{array}\right), 0,0,0,0,0, \ldots} \\\ {\text { f) } 2,-2,2,-2,2,-2,2,-2, \ldots}\end{array} $$ $$ \begin{array}{l}{\text { g) } 1,1,0,1,1,1,1,1,1,1,1, \ldots} \\ {\text { h) } 0,0,0,1,2,3,4, \ldots}\end{array} $$

Step by step solution

01

Understand the Problem

We need to find a closed form (an algebraic expression not using summation or ellipses) for each sequence.

02

Identify Patterns and Formulas

Before finding the generating function, observe the pattern in each sequence.

03

Sequence Analysis and Generating Function Formulation

Formulate the algebraic expression for each sequence and derive the generating function.

04

Sub-step 3(a): Analyze Sequence (a)

Sequence: \(0,2,2,2,2,2,2,0,0,0,0,0, \ldots\)\ Formula: This sequence can be written as \(2x + 2x^2 + 2x^3 + 2x^4 + 2x^5 + 2x^6\). So the generating function is \(2(x + x^2 + x^3 + x^4 + x^5 + x^6) = 2x\frac{1-x^6}{1-x} = 2x-2x^7\).

05

Sub-step 3(b): Analyze Sequence (b)

Sequence: \(0,0,0,1,1,1,1,1,1, \ldots\)\ Formula: This sequence can be written as \(x^3 + x^4 + x^5 + x^6 + x^7\). The generating function is \(x^3\frac{1-x^5}{1-x} = x^3-x^8\).

06

Sub-step 3(c): Analyze Sequence (c)

Sequence: \(0,1,0,0,1,0,0,1,0,0,1, \ldots\) \ Formula: This sequence can be written as \(x + x^4 + x^7 + x^{10} + \ldots\). So the generating function is \(x(1 + x^3 + x^6 + \ldots) = x\frac{1}{1-x^3}\).

07

Sub-step 3(d): Analyze Sequence (d)

Sequence: \(2,4,8,16,32,64,128,256, \ldots\)\ Formula: This sequence can be written as \(2^1, 2^2, 2^3, 2^4, 2^5, \ldots\). The generating function is \(\sum_{n=0}^\infty 2^n x^n = \frac{1}{1-2x}\).

08

Sub-step 3(e): Analyze Sequence (e)

Sequence: \(\binom{7}{0}, \binom{7}{1}, \binom{7}{2}, \ldots , \binom{7}{7}, 0,0,0,0,0, \ldots\)\ Formula: This sequence can be represented by the binomial coefficient. The closed form for generating function of this sequence is \((1+x)^7\).

09

Sub-step 3(f): Analyze Sequence (f)

Sequence: \(2,-2,2,-2,2,-2,2,-2, \ldots\)\ Formula: This sequence can be written as \(2(1 -x + x^2 - x^3 + x^4 - x^5 + \ldots)\). The generating function is \(\frac{2}{1+x}\).

10

Sub-step 3(g): Analyze Sequence (g)

Sequence: \(1,1,0,1,1,1,1,1,1,1,1, \ldots\) \ Formula: This sequence can be written as \(1 + x + x^3 + x^4 + x^5 + x^6 + \ldots\). The generating function is \(1 + x + x^3 + x^4 + x^5 \ldots = \frac{1+x+x^3}{1-x}\).

11

Sub-step 3(h): Analyze Sequence (h)

Sequence: \(0,0,0,1,2,3,4, \ldots\)\ Formula: This sequence can be written as \(x^3 (1 + 2x + 3x^2 + 4x^3 \ldots )\). The generating function is \(\frac{x^3}{(1-x)^2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

closed form

In mathematics, a closed form expression is an algebraic expression that does not use summation (Σ) or ellipses (...). For instance, if you have a series or sequence, representing it in a closed form means finding a finite expression that encapsulates the entire series. For example, instead of writing out infinite terms or showing the sum of many elements individually, you compact it into a single, simplified formula. This closed form makes it easier to analyze, compute, and understand sequences.

sequence patterns

Detecting patterns in sequences is essential for finding their closed forms. A sequence pattern helps to predict the next terms and allows mathematicians to compactly represent the entire series using a generating function. For example, in the sequence of the powers of 2 \(2, 4, 8, 16, \ldots\), we can recognize the pattern of each term being double the previous one. Recognizing such patterns helps in writing the sequence in a mathematical form, making further calculations or analysis straightforward.

algebraic expression

An algebraic expression is a combination of constants, variables, and operations (addition, subtraction, multiplication, division, and exponentiation). When we convert sequences into their closed forms, we use algebraic expressions to represent the relationships or patterns within the sequences. For example, consider the sequence \[0,1,0,0,1,0,0,1,0,0,1,\ldots \]. We can represent this using the algebraic expression \((x + x^4 + x^7 +\ldots = x\frac{1}{1-x^3}) \). These algebraic expressions help simplify complex series into manageable forms.

discrete mathematics

Discrete mathematics is the branch of mathematics dealing with discrete elements that use distinct, separate values. This is opposed to continuous mathematics which deals with objects that can vary smoothly. Sequences and generating functions are fundamental parts of discrete mathematics since they deal with countable, distinct objects. Through studying discrete mathematics, we learn specific methods to analyze sequences, detect patterns, and produce generating functions. Overall, discrete mathematics helps understand structures that are fundamentally discrete and thus frequently found in computer science, information theory, and combinatorics.