/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 What is the general form of the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 22

What is the general form of the solutions of a linear homogeneous recurrence relation if its characteristic equation has the roots \(-1,-1,-1,2,2,5,5,7 ?\)

Short Answer

Expert verified
The general solution is \[a_n = c_1 (-1)^n + c_2 n(-1)^n + c_3 n^2(-1)^n + c_4 2^n + c_5 n 2^n + c_6 5^n + c_7 n 5^n + c_8 7^n\].

Step by step solution

01

Identify the Roots

The roots of the characteristic equation are \(-1, -1, -1, 2, 2, 5, 5, 7\).
02

Determine the Form for Repeated Roots

For each root with multiplicity, we need to include terms for each repetition. For example, for a root \(-1\) with multiplicity 3, the terms will be \(c_1 (-1)^n, c_2 n(-1)^n, c_3 n^2(-1)^n\).
03

Write the Solution for Each Root

The terms for each root are as follows: \(c_1 (-1)^n, c_2 n(-1)^n, c_3 n^2(-1)^n\) for the root \(-1\) with multiplicity 3; \(c_4 2^n, c_5 n 2^n\) for the root \(2\) with multiplicity 2; \(c_6 5^n, c_7 n 5^n\) for the root \(5\) with multiplicity 2; and \(c_8 7^n\) for the root \(7\) with multiplicity 1.
04

Combine All Terms

Combine all the terms obtained from each root: \[a_n = c_1 (-1)^n + c_2 n(-1)^n + c_3 n^2 (-1)^n + c_4 2^n + c_5 n 2^n + c_6 5^n + c_7 n 5^n + c_8 7^n\].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Characteristic Equation
In the study of linear homogeneous recurrence relations, the characteristic equation is a crucial concept.
It's a polynomial obtained from the recurrence relation that helps determine the form of its solutions.
For a linear homogeneous recurrence relation of order 'k': a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_k a_{n-k}
we derive its characteristic equation by assuming solutions in the form of powers, subbing back in, and simplifying. This characteristic polynomial looks like: a_n = c_1 r^{n-1} + c_2 r^{n-2} + ... + c_k r^{n-k} = 0.
By solving this polynomial, we find the roots, which are crucial for building the general solution.
Roots Multiplicity
Roots multiplicity occurs when a root of the characteristic equation appears more than once.
For instance, if a root 'r' appears 3 times, it has a multiplicity of 3.
This means that the root contributes multiple terms to the general solution. Each repeated root must be paired with increasingly higher powers of 'n' to form distinct solutions.
For a root 'r' with multiplicity 'm', the corresponding terms in the general solution will be r^n, n*r^n, n^2*r^n, ..., n^{m-1}*r^n.
In our example, the roots equation: r = -1 (multiplicity 3)r = 2 (multiplicity 2) r = 5 (multiplicity 2)r = 7 (multiplicity 1) are used accordingly.
General Solution Form
The general solution form of a linear homogeneous recurrence relation is a combination of terms derived from the roots of its characteristic equation.
For each unique root 'r' with multiplicity 'm', we include 'm' terms in the general solution.
Each term is a product of a constant coefficient and a combination of 'r' raised to increasing powers multiplied by powers of 'n'.
For example, in our specific case, the roots from the characteristic equation give us this general solution: a_n = c_1 (-1)^n + c_2 n(-1)^n + c_3 n^2 (-1)^n + c_4 2^n + c_5 n 2^n + c_6 5^n + c_7 n 5^n + c_8 7^n
Recurrence Relations
Recurrence relations are equations that recursively define a sequence.
Each term is formulated as a function of its predecessor terms.
A linear homogeneous recurrence relation has the form: a_n = c_1 a_{n-1} + c_2 a_{n-2} + ... + c_k a_{n-k},
where the sequence {a_n} is dependent purely on the terms that came before, and is called 'homogeneous' because there is no additional function of 'n' on the right hand side.
Understanding the characteristic equation, roots multiplicities, and general solution forms is crucial to solving these recurrence relations efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

a) Find a recurrence relation for the number of ternary strings of length \(n\) that do not contain consecutive symbols that are the same. b) What are the initial conditions? c) How many ternary strings of length six do not contain consecutive symbols that are the same?

Suppose that the function \(f\) satisfies the recurrence relation \(f(n)=2 f(\sqrt{n})+1\) whenever \(n\) is a perfect square greater than 1 and \(f(2)=1\) a) Find \(f(16)\) . b) Give a big- \(O\) estimate for \(f(n) .\) [Hint: Make the substitution \(m=\log n . ]\)

Find the coefficient of \(x^{9}\) in the power series of each of these functions. a) \(\left(1+x^{3}+x^{6}+x^{9}+\cdots\right)^{3}\) b) \(\left(x^{2}+x^{3}+x^{4}+x^{5}+x^{6}+\cdots\right)^{3}\) c) \(\left(x^{3}+x^{5}+x^{6}\right)\left(x^{3}+x^{4}\right)(x+\cdots)^{3}\) d) \(\left(x+x^{4}+x^{7}+x^{10}+\cdots\right)\left(x^{2}+x^{4}+x^{6}+x^{8}+\cdots\right)\) e) \(\left(1+x+x^{2}\right)^{3}\)

Generating functions are useful in studying the number of different types of partitions of an integer \(n .\) A partition of a positive integer is a way to write this integer as the sum of positive integers where repetition is allowed and the or- der of the integers in the sum does not matter. For exam- ple, the partitions of 5 (with no restrictions) are \(1+1+1+1+\) \(1+1,1+1+1+2,1+1+3,1+2+2,1+4,2+3,\) and \(5 .\) Exercises \(53-58\) illustrate some of these uses. Show that the coefficient \(p_{d}(n)\) of \(x^{n}\) in the formal power series expansion of \((1+x)\left(1+x^{2}\right)\left(1+x^{3}\right) \cdots\) equals the number of partitions of \(n\) into distinct parts, that is, the number of ways to write \(n\) as the sum of positive inte- gers, where the order does not matter but no repetitions are allowed.

Use generating functions to solve the recurrence relation \(a_{k}=5 a_{k-1}-6 a_{k-2}\) with initial conditions \(a_{0}=6\) and \(a_{1}=30\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.