/*! 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 58 Exercises 58 and 59 refer to the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 58

Exercises 58 and 59 refer to the sequence \(S_{n}\) defined by $$ S_{1}=0, \quad S_{2}=1, \quad S_{n}=\frac{S_{n-1}+S_{n-2}}{2} \quad n=3,4, \ldots $$ Compute \(S_{3}\) and \(S_{4}\).

Short Answer

Expert verified
\(S_{3}=0.5\) and \(S_{4}=0.75\)

Step by step solution

01

Compute S_{3}

The formula for determining \(S_{n}\) when \(n=3, 4, \ldots\) is \(S_{n}=\frac{S_{n-1}+S_{n-2}}{2}\). Plugging in 3 as \(n\) gives us \(S_{3}=\frac{S_{3-1}+S_{3-2}}{2} = \frac{S_{2}+S_{1}}{2}\). We know from the exercise that \(S_{2}=1\) and \(S_{1}=0\). Plugging these values gives us \(S_{3}=\frac{1+0}{2} = 0.5\)
02

Compute S_{4}

To find \(S_{4}\), we use the same formula \(S_{n}=\frac{S_{n-1}+S_{n-2}}{2}\). Plugging in 4 as \(n\) gives us \(S_{4}=\frac{S_{4-1}+S_{4-2}}{2} = \frac{S_{3}+S_{2}}{2}\). We have acquired from the previous steps that \(S_{3}=0.5\) and \(S_{2}=1\). Plugging these values gives us \(S_{4}=\frac{0.5+1}{2} = 0.75\)

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.

Recurrence Relations
A recurrence relation is a way of defining sequences based on a rule that derives each term from the ones preceding it. This can help in efficiently determining future values without needing to compute every previous term each time.

For example, consider the sequence given by:
  • \( S_1 = 0 \)
  • \( S_2 = 1 \)
  • \( S_n = \frac{S_{n-1} + S_{n-2}}{2} \) for \( n = 3, 4, \ldots \)
To calculate any term \( S_n \), you need only the two preceding values, \( S_{n-1} \) and \( S_{n-2} \). This dependency makes recurrence relations powerful for sequences where the pattern is defined by previous terms. For instance, to find \( S_3 \) and \( S_4 \), like in the exercise:
  • \( S_3 = \frac{S_2 + S_1}{2} = \frac{1 + 0}{2} = 0.5 \)
  • \( S_4 = \frac{S_3 + S_2}{2} = \frac{0.5 + 1}{2} = 0.75 \)
Arithmetic Mean
The arithmetic mean is a way to find the average of numbers. It鈥檚 a basic mathematical concept often used in statistics and number sequences.

In the context of the given sequence, each new term is defined using the arithmetic mean of the two preceding terms. The formula for the arithmetic mean is\[\text{Arithmetic Mean} = \frac{\text{Sum of the Numbers}}{\text{Count of Numbers}}\]So, for two numbers \(a\) and \(b\), the mean is:\[\frac{a + b}{2}\] Using this, to find \( S_3 \) with the terms \( S_2 \) and \( S_1 \), compute:
  • \( S_3 = \frac{1 + 0}{2} = 0.5 \)
Similarly, to find \( S_4 \):
  • \( S_4 = \frac{0.5 + 1}{2} = 0.75 \)
Mathematical Induction
Mathematical induction is a technique primarily used to prove that a statement is true for all natural numbers. It鈥檚 akin to a chain reaction, where if one link is true, all links in the sequence are considered true.

While not directly applied in the original exercise, understanding mathematical induction can deepen your comprehension of how to prove sequences systematically. Induction generally involves two steps:
  • Base Step: Establish the statement for an initial value, often \( n = 1 \).
  • Inductive Step: Assume the statement is true for \( n = k \), and then prove it for \( n = k+1 \).
For the sequence \( S_n \), you could use induction to show that this recursive method always leads to defined values for all \( n \) starting from \( n=3 \). It's an effective way to assure correctness and the pattern of results across potentially infinite terms.

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

Write a \(\Theta(n \lg n)\) algorithm that finds the distance \(\delta\) between a closest pair and all pairs less than \(2 \delta\) apart, where each pair less than \(2 \delta\) apart is listed exactly one time.

List all inversions in the permutation 1,2,3,4 .

Explain why the following algorithm does not find the distance \(\delta\) between a closest pair and all pairs less than \(2 \delta\) apart. exercise \(14(p, n)\) $$ \begin{array}{l} \delta=\text { closest-pair }(p, n) \\ \text { for } i=1 \text { to } n-1 \\ \text { for } j=i+1 \text { to } \min \\{n, i+31\\} \\ \text { if }\left(\operatorname{dist}\left(p_{i}, p_{j}\right)<2 * \delta\right) \\ \quad \operatorname{println}(i+" "+j) \end{array} $$

A network consists of \(n\) nodes. Each node has communications facilities and local storage. Periodically, all files must be shared. A link consists of two nodes sharing files, Specif. ically, when nodes \(A\) and \(B\) are linked, \(A\) transmits all its files to \(B\) and \(B\) transmits all its files to \(A\). Only one link exists at a time, and after a link is established and the files are shared, the link is deleted. Let \(a_{n}\) be the minimum number of links required by \(n\) nodes so that all files are known to all nodes. (a) Show that \(a_{2}=1, a_{3} \leq 3, a_{4} \leq 4\). (b) Show that \(a_{n} \leq a_{n-1}+2, n \geq 3\).

In Exercises \(47-52,\) Ackermann's original fiuncrion \(A O\) is defined by $$ \begin{array}{cc} A O(0, y, z)=z+1 & y, z \geq 0 \\ A O(1, y, z)=y+z & y, z \geq 0 \\ A O(2, y, z)=y z & y, z \geq 0 \\ A O(x+3, y, 0)=1 & x \geq 0, y \geq 1, \\ A O(x+3, y, z+1)= & \\ A O(x+2, y, A O(x+3, y, z)) & x \geq 0, y \geq 1, z \geq 0 . \end{array} $$ Exercises \(47-52\) refer to the function \(A O\) and to Ackermann's function \(A\). $$ \text { Prove that } A O(3, y, z)=y^{2} \text { for } y \geq 1, z \geq 0 \text { . } $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.