/*! 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 8 Determine the number of \(n\)-di... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 8

Determine the number of \(n\)-digit quaternary \((0,1,2,3)\) sequences in which there is never a 3 anywhere to the right of a 0 .

Short Answer

Expert verified
To determine total number of \(n\)-digit quaternary sequences where there is never a 3 anywhere to the right of a 0, calculate number of valid sequences for each case and then add them up. Use the formulas \(a_{n}=a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1}\), \(b_{n}=a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1}\), \(c_{n}=a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1}\), and \(d_{n}=a_{n-1}+b_{n-1}+d_{n-1}\) iterated from base cases, until the desired term is reached. Then add \(a_n + b_n + c_n + d_n\).

Step by step solution

01

Analyze the condition

Before starting, you should understand that having a 3 anywhere to the right of a 0 is not allowed. This condition means even 01, 02 and 03 can be formed, any combination ending with 0 can only be followed by 0,1 and 2 but never by 3. This restriction limits the possible quaternary numbers that can be produced.
02

Split into Cases

Given the restrictions, we can split our problem into four cases considering the end number of sequences by 0,1,2 and 3. If a sequence ends with 1,2, or 3, there are no limitations for previous numbers; however for the ones ending with 0, they can only be followed by sequences ending with 0,1 and 2 because of our condition which is not having 3 anywhere to the right of 0. Consequently, we will have 4 cases which can be described as sequences ending with 0,1,2,3.
03

Count the number of sequences for each case

Let's denote the number of n-digit sequences ending with 1,2,3, and 0 as \(a_n\), \(b_n\), \(c_n\), and \(d_n\) respectively. For a sequence to end in 1, 2, or 3, the previous sequence could be any \(n-1\) digit sequence, thus \(a_{n}=a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1}\), \(b_{n}=a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1}\), and \(c_{n}=a_{n-1}+b_{n-1}+c_{n-1}+d_{n-1}\). However, for a sequence to end in 0, the preceding sequence could not end in 3. Thus, \(d_{n}=a_{n-1}+b_{n-1}+d_{n-1}\).
04

Find the base cases

Observe that for the base case when \(n = 1\), \(a_1\), \(b_1\), \(c_1\), and \(d_1\) are all 1, since each of '1', '2', '3', and '0' are 1-digit quaternary number.
05

Iterate upon base cases to reach desired term

Applying the formulas derived from Step 3 and iterating from the base cases, you can get to the required term by applying the formulas repeatedly until \(n\) terms are reached.
06

Solution

Finally, add \(a_n\), \(b_n\), \(c_n\), and \(d_n\) to get the number of all admissible sequences. The final solution is \(a_n + b_n + c_n + d_n\).

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!

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

In Exercise 14 of Section \(4.2\) we learned that \(F_{0}+F_{1}+F_{2}+\cdots+F_{n}=\sum_{i=0}^{n} F_{l}=F_{n+2}-1\). This is one of many such properties of the Fibonacci numbers that were discovered by the French mathematician Fran莽ois Lucas (1842-1891). Although we established the result by mathematical induction, we see that it is easy to develop this formula by adding the system of \(n+1\) equations $$ \begin{gathered} F_{0}=F_{2}-F_{1} \\ F_{1}=F_{3}-F_{2} \\ \cdots \\ F_{n-1}=F_{n+1}-F_{n} \\ F_{n}=F_{n+2}-F_{n+1} . \end{gathered} $$ Develop formulas for each of the following sums, and then check the general result by mathematical induction. a) \(F_{1}+F_{3}+F_{5}+\cdots+F_{2 n-1}\), where \(n \in \mathbf{Z}^{+}\) b) \(F_{0}+F_{2}+F_{4}+\cdots+F_{2 n}\), where \(n \in \mathbf{Z}^{+}\)

a) For \(\alpha=(1+\sqrt{5}) / 2\), verify that \(\alpha^{2}+1=2+\alpha\) and \((2+\alpha)^{2}=5 \alpha^{2}\). b) Show that for \(\beta=(1-\sqrt{5}) / 2, \beta^{2}+1=2+\beta\) and \((2+\beta)^{2}=5 \beta^{2}\). c) If \(n, m \in \mathbf{N}\) prove that \(\sum_{k-0}^{2 n}\left(\begin{array}{c}2 n \\ k\end{array}\right) F_{2 k+m}=\) \(5^{n} F_{2 n+m}\).

On the first day of a new year, Joseph deposits \(\$ 1000\) in an account that pays \(6 \%\) interest compounded monthly. At the beginning of each month he adds \(\$ 200\) to his account. If be continues to do this for the next four years (so that he makes 47 additional deposits of \(\$ 200\) ), how much will his account be worth exactly four years after he opened it?

Solve the following recurrence relations. a) \(a_{n+2}^{2}-5 a_{n+1}^{2}+6 a_{n}^{2}=7 n, \quad n \geq 0, \quad a_{0}=a_{1}=1\) b) \(a_{n}+n a_{n-1}=n !, n \geq 1, a_{0}=1\) c) \(a_{n}^{2}-2 a_{n-1}=0, \quad n \geq 1, a_{0}=2\) (Let \(b_{n}=\log _{2} a_{n}, n \geq 0\).)

Let \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) be \(n\) real numbers, with \(x_{1} \leq x_{2} \leq x_{3} \leq \cdots \leq x_{n}\). The median for this set of \(n\) numbers is defined as $$ \begin{aligned} &x_{(n+1) / 2}, \text { for } n \text { odd } \\ &(1 / 2)\left[x_{n / 2}+x_{(n / 2)+1}\right], \text { for } n \text { even. } \end{aligned} $$ a) Suppose that \(A[1], A[2], A[3], \ldots, A[n]\) is an array of \(n\) real numbers, not necessarily in ascending or descending order. Write a computer program (or develop an algorithm) to determine the median for the set of real numbers listed in this array. b) Discuss the worst-case time complexity for the program you wrote in part (a).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.