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Chapter 7: Problem 12

Find functions defined on the set of nonnegative integers that define the sequences whose first six terms are given below. a. \(1,-\frac{1}{3}, \frac{1}{5},-\frac{1}{7}, \frac{1}{9},-\frac{1}{11}\) b. \(0,-2,4,-6,8,-10\)

Short Answer

Expert verified
The functions that define the sequences for the given problems are: a. \(f(n) = \frac{(-1)^n}{(2n+1)}\) b. \(f(n) = (-1)^n (2n)\)

Step by step solution

01

The pattern seems to alternate between positive and negative numbers. The denominators of the fractions are increasing odd integers. #Step 2: Formulate a function#

We need a function that outputs positive and negative numbers, so we will use the function \((-1)^n\). This will alternate between 1 and -1 as n increases. Now we need to multiply this by the increasing odd integers as denominators. #Step 3: Create a function for increasing odd integers#
02

As n increases, we can generate a sequence of odd integers with the formula: \(2n + 1\). For example, when n = 0, 1, 2, 3, ... the output will be 1, 3, 5, 7, ... #Step 4: Combine the functions#

Now we can combine our functions to create our final function: \[f(n) = \frac{(-1)^n}{(2n+1)}\] #Step 5: Check the function against the given terms#
03

To confirm if our function works, we can input the first six nonnegative integers (0, 1, 2, 3, 4, 5) and see if we get the given sequence. b. \(0, -2, 4, -6, 8, -10\) #Step 1: Analyze the pattern of the sequence#

The pattern seems to alternate between positive and negative numbers. The magnitudes of the values are increasing even integers. #Step 2: Formulate a function#
04

We need a function that outputs positive and negative numbers, similar to the first example. We can use the function \((-1)^n\). Now, we need to multiply this by the increasing even integers. #Step 3: Create a function for increasing even integers#

As n increases, we can generate a sequence of even integers with the formula: \(2n\). For example, when n = 0, 1, 2, 3, ... the output will be 0, 2, 4, 6, ... #Step 4: Combine the functions#
05

Now we can combine our functions to create our final function: \[f(n) = (-1)^n (2n)\] #Step 5: Check the function against the given terms#

To confirm if our function works, we can input the first six nonnegative integers (0, 1, 2, 3, 4, 5) and see if we get the given sequence.

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Key Concepts

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

Function Formulation
In discrete mathematics, creating precise functions based on a sequence of numbers is an important skill. Function formulation involves crafting mathematical expressions that define the operations needed to generate a sequence from a set of inputs, often with the input being the set of nonnegative integers.

For example, in the sequences provided, the functions must account for alternating signs and changing denominators or multipliers that are evident in the sequences' structure. By identifying patterns and properties within the sequences, such as alternating signs or progression of odd and even integers, we define functions that reproduce the given terms. This encapsulates the process of function formulation, converting observed patterns into algebraic expressions.
Odd and Even Integers
Understanding odd and even integers is fundamental in the analysis of numerical sequences. An even integer is divisible by 2, while an odd integer is not; it has a remainder of 1 when divided by 2.

Sequences often feature these numbers, and recognizing their properties can be crucial in function formulation. For increasing even integers, we use the formula \(2n\), whereas for increasing odd integers, the formula \(2n+1\) is used, where \(n\) represents the elements in the sequence starting from zero, or another initial integer. The ability to generate odd and even integers using these expressions is a cornerstone in constructing mathematical sequences and is essential for solving related problems in discrete mathematics.
Sequence Pattern Analysis
Sequence pattern analysis involves identifying the systematic components that dictate the formation of a sequence. Key to discovering these patterns is looking at the regularity in the sequence's progression, such as alternating signs or arithmetic changes in the values.

For instance, analyzing the sequence \(1, -\frac{1}{3}, \frac{1}{5}, -\frac{1}{7}, \ldots\), the pattern of alternating positive and negative fractions and the continuous sequence of increasing odd denominators become apparent. Observing the regularity in change鈥攂e it through numerical intervals, signs, or other properties鈥攁llows for the construction of a general formula that can predict any term in the sequence. Effective pattern analysis makes the difference between solving a sequence problem and being stumped by it.
Mathematical Induction
Mathematical induction is a powerful proof technique used to establish the truth of an infinite number of cases, often involving sequences or series. The process begins with proving the base case, the initial term of a sequence. Then, assuming that a statement holds for some arbitrary case \(n\), we prove that it must also hold for the next case \(n+1\), demonstrating the truth of the statement for all cases following the base.

When applied to our sequence problem, we could use mathematical induction to prove that the functions formulated indeed generate the intended sequences for all nonnegative integer inputs. The process would involve showing that the functions work for the first term and that if they work for an arbitrary term \(n\), they must also work for the term \(n+1\). This approach doesn't just demonstrate a pattern; it provides a rigorous foundation for the function's validity across the entire sequence.

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Most popular questions from this chapter

The functions of each pair in \(9-11\) are inverse to each other. For each pair, check that both compositions give the identity function. \(F: \mathbf{R} \rightarrow \mathbf{R}\) and \(F^{-1}: \mathbf{R} \rightarrow \mathbf{R}\) are defined by \(F(x)=3 x+2, \quad\) for all \(x \in \mathbf{R}\) and \(F^{-1}(y)=\frac{y-2}{3}, \quad\) for all \(y \in \mathbf{R}\)

Let \(S\) be the set of all strings of 0 's and 1 's, and define \(l: S \rightarrow \mathbf{Z}^{\text {nonneg }}\) by \(l(s)=\) the length of \(s\), for all strings \(s\) in \(S .\) a. Is \(l\) one-to-one? Prove or give a counterexample. b. Is \(l\) onto? Prove or give a counterexample.

In a group of 30 people, must at least 3 have been born in the same month? Why?

Let \(S\) be the set of all strings of \(a\) 's and \(b^{\prime}\) 's. a. Define \(f: S \rightarrow \mathbf{Z}\) as follows: For each string \(s\) in \(f(s)=\left\\{\begin{array}{l}\text { the number of } b \text { 's to the left } \\ \text { of the left-most } a \text { in } s \\ 0 \quad \text { if } s \text { contains no } a \text { 's }\end{array}\right.\) Find \(f(a b a), f(b b a b)\), and \(f(b)\). What is the range of \(f\) ? b. Define \(g: S \rightarrow S\) as follows: For each string \(s\) in \(S\), $$ g(s)=\text { the string obtained by writing the } $$ characters of \(s\) in reverse order. Find \(g(a b a), g(b b a b)\), and \(g(b)\). What is the range of \(g\) ?

Exercises \(40-47\) refer to the following definition: Definition: If \(f: X \rightarrow Y\) is a function and \(A \subseteq X\) and \(C \subseteq Y\) then $$ f(A)=\\{y \in Y \mid y=f(x) \text { for some } x \text { in } A\\} $$ and $$ f^{-1}(C)=\\{x \in X \mid f(x) \in C\\} $$ Determine which of the properties in \(40-47\) are true for all functions \(f\) from a set \(X\) to a set \(Y\) and which are false for some function \(f\). Justify your answers. For all subsets \(C\) and \(D\) of \(Y\), $$ f^{-1}(C-D)=f^{-1}(C)-f^{-1}(D) $$
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