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Chapter 6: Problem 9

Solve the recurrence relation \(d_{n}=-d_{n-1}, n \geq 2,\) where \(d_{1}=-1\).

Short Answer

Expert verified
The general formula for the given recurrence relation \(d_n = -d_{n-1}\), with the initial condition \(d_1 = -1\), is given by: \[ d_n = \begin{cases} -1 & \text{if } n \text{ is odd} \\ 1 & \text{if } n \text{ is even} \end{cases} \] This formula holds for all values of \(n \geq 2\) and satisfies the initial condition.

Step by step solution

01

Observe the pattern

Notice that the given recurrence relation alternates the sign of the terms. It's easy to see that the first few terms are: \(d_1 = -1\), \(d_2 = -(-1) = 1\), \(d_3 = -(1) = -1\), \(d_4 = -(-1) = 1\), and we can keep going, applying the rule.
02

Obtain the general formula

Observe that the sign alternates between each term, with odd indices being negative and even indices being positive. Based on this pattern, we can derive the following general formula for \(d_n\): \[ d_n = \begin{cases} -1 & \text{if } n \text{ is odd} \\ 1 & \text{if } n \text{ is even} \end{cases} \]
03

Verify the general formula

To see if the general formula holds, let's try a few values of \(n\) and compare them to the pattern we observed in Step 1: For \(n=1\), \(d_1 = -1\) since 1 is odd. For \(n=2\), \(d_2 = 1\) since 2 is even. For \(n=3\), \(d_3 = -1\) since 3 is odd. For \(n=4\), \(d_4 = 1\) since 4 is even. It appears that the general formula correctly gives the values of \(d_n\) for the tested values of \(n\).
04

Conclude

From our analysis, we have found the general formula for the given recurrence relation: \[ d_n = \begin{cases} -1 & \text{if } n \text{ is odd} \\ 1 & \text{if } n \text{ is even} \end{cases} \] This formula holds for all values of \(n \geq 2\), and it satisfies the initial condition \(d_1 = -1\).

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Key Concepts

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

Alternating Sequence
An alternating sequence is a mathematical pattern in which each successive term differs from the previous one by a sign change. For the sequence in question, we viewed the recurrence relation, where each term is the negation of the term before it. To better understand this concept, consider flipping a light switch on and off repeatedly; the state of the light alternates with each flip, akin to how the terms in our sequence alternate between positive and negative values.

Typically, an alternating sequence can be expressed in the form \( (-1)^n \times a_n \), where \( a_n \) is a sequence of numbers and \( n \) indicates the term's position. This characteristic is essential in identifying and solving problems associated with alternating sequences in discrete mathematics.
General Formula
The general formula of a sequence provides an expression that can determine any term's value based on its position. It encapsulates the pattern of a sequence in a single, definitive equation, allowing us to find, for example, the 50th term without calculating all preceding 49 terms. This power of expression simplifies computations and illuminates the structure of the sequence.

For the given sequence, the formula is defined piecewise鈥攄ifferent expressions handle different cases (odd or even positions). This type of general formula is particularly effective for alternating sequences, where one might encounter starkly contrasting behavior for consecutive terms.
Mathematical Induction
The concept of mathematical induction is a method of proof used to establish the truth of an infinite sequence of statements. It's akin to a line of dominoes falling; if the first domino falls, and each domino is guaranteed to knock down the next, then all the dominoes will eventually fall. Applied to the sequence in our problem, induction could help us prove that our general formula holds true for every term after our base case.

To perform mathematical induction, we start by proving that the initial statement (often called the base case) is true. Then, we assume a typical case, called the induction hypothesis, and prove that if this is true for a particular instance, it will be true for the next instance (the inductive step). This creates a chain reaction proving that the statement is true for all natural numbers.
Discrete Mathematics
Discrete mathematics is a branch of mathematics that deals with discrete elements, often dealing with countable, non-continuous structures. It encompasses a variety of topics including logic, set theory, graph theory, and combinatorics. Discrete mathematics is the bedrock of computer science and informs theories and practical applications alike.

In discrete mathematics, recurrence relations and sequences play a crucial role in modeling and solving problems that involve discrete steps or stages. Understanding the characteristics of alternating sequences, developing general formulas, and mastery in techniques like mathematical induction are fundamental skills for anyone delving into this field.

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Most popular questions from this chapter

A survey shows that \(20 \%\) of the adults in Simpleton have high blood pressure. A sample of four adults is selected at random. Find the probability that: Not all of them have high blood pressure.

Using the binomial theorem, prove each. \(\sum_{r=0}^{n}\left(\begin{array}{c}n \\\ r\end{array}\right)\left(\begin{array}{c}n \\\ n-r\end{array}\right)\left(\begin{array}{c}2 n \\ n\end{array}\right)\) [Hint: Consider \((1+x)^{2 n}=(1+x)^{n}(1+x)^{n} .\) Equate the coefficients of \(\left.x^{n} \text { from either side. }\right]\)

Using induction, prove each. $$\left(\begin{array}{l}n \\ 0\end{array}\right)^{2}+\left(\begin{array}{l}n \\\ 1\end{array}\right)^{2}+\left(\begin{array}{l}n \\\ 2\end{array}\right)^{2}+\cdots+\left(\begin{array}{l}n \\\ n\end{array}\right)^{2}=\left(\begin{array}{l}2 n \\ n\end{array}\right)$$

Evaluate each sum. \(1\left(\begin{array}{l}n \\ 1\end{array}\right)+2\left(\begin{array}{l}n \\\ 2\end{array}\right)+3\left(\begin{array}{l}n \\\ 3\end{array}\right)+\cdots+n\left(\begin{array}{l}n \\ n\end{array}\right)\) (Hint: Let \(S\) denote the sum. Use \(S\) and the sum in the reverse order to compute \(2 S .\))

A survey conducted among 300 adults show that 160 like to have their houses painted white and 140 like blue, Seventy-Four like both colors. How do not like either colors?
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