/*! 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 20 Write the first four terms of th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 20

Write the first four terms of the sequence \(\left\\{a_{n}\right\\}\) defined by the following recurrence relations. $$a_{n+1}=a_{n}^{2}-1 ; a_{1}=1$$

Short Answer

Expert verified
Question: Determine the first four terms of the sequence \(\left\\{a_{n}\right\\}\) defined by the recurrence relation \(a_{n+1}=a_{n}^{2}-1\) with the initial term \(a_1 = 1\). Answer: The first four terms of the sequence are: \(a_1 = 1,\ a_2 = 0,\ a_3 = -1,\ a_4 = 0\).

Step by step solution

01

Find the first term of the sequence

First, we are given the first term of the sequence, \(a_1\). The problem states that \(a_1 = 1\). So, the first term of the sequence \(\left\\{a_{n}\right\\}\) is: $$a_1 = 1$$
02

Find the second term of the sequence

To find the second term of the sequence, \(a_2\), we will use the given recurrence relation \(a_{n+1}=a_{n}^{2}-1\). We plug in n=1 and use the first term \(a_1\) in the recurrence relation: $$a_2 = a_{1}^{2} - 1 = 1^2 - 1 = 0$$ So, the second term of the sequence \(\left\\{a_{n}\right\\}\) is: $$a_2 = 0$$
03

Find the third term of the sequence

Now, we will find the third term of the sequence, \(a_3\), by plugging in n=2 and using the second term \(a_2\) in the recurrence relation: $$a_3 = a_{2}^{2} - 1 = 0^2 - 1 = -1$$ So, the third term of the sequence \(\left\\{a_{n}\right\\}\) is: $$a_3 = -1$$
04

Find the fourth term of the sequence

Lastly, we will find the fourth term of the sequence, \(a_4\), by plugging in n=3 and using the third term \(a_3\) in the recurrence relation: $$a_4 = a_{3}^{2} - 1 = (-1)^2 - 1 = 0$$ So, the fourth term of the sequence \(\left\\{a_{n}\right\\}\) is: $$a_4 = 0$$
05

Final Answer

The first four terms of the sequence \(\left\\{a_{n}\right\\}\) defined by the given recurrence relation are: $$a_1 = 1,\ a_2 = 0,\ a_3 = -1,\ a_4 = 0$$

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.

Sequences in Calculus
A sequence in calculus is an ordered list of numbers, typically represented by a formula or rule that defines the position of each number in the list. These sequences can have a variety of applications, such as modeling natural phenomena or representing series expansions in analysis.

Each number in the sequence is called a term, and each term is defined by its position, denoted by a subscript. For example, in the sequence \( \(a_n\) \) the term \( a_1 \) is the first term, \( a_2 \) is the second term, and so on. Sequences can be either finite, having a specific number of terms, or infinite, continuing indefinitely.

When working with sequences, it's essential to understand the underlying pattern or rule that generates the terms. This rule can be explicit, wherein each term is defined independently, or recursive, where each term is defined in terms of previous terms, like in the given exercise.
Recurrence Relation Calculus
Recurrence relations are a type of equation that defines the terms of a sequence using the values of previous terms. In the field of calculus and discrete mathematics, these relations are particularly vital for modeling progression where the next state depends on the current one.

A simple linear recurrence relation looks like \( a_{n+1}=f(a_n) \), where the function \( f \) operates on the \( n \)th term to produce the next term in the sequence. In more complex forms, a recurrence relation can involve several previous terms and even coefficients that might vary with \( n \).

Recurrence relations require initial conditions to be fully defined. These are the starting values from which the entire sequence can be determined. In the earlier example, \( a_1 \) serves as this initial value, giving us a starting point to calculate subsequent terms.
Sequence Terms Calculation
Calculating terms of a sequence defined by a recurrence relation involves iterating the defined rule starting from the initial terms. This process is crucial for pattern recognition, numerical analysis, and even algorithm design in computer science.

In our exercise, after establishing the first term \( a_1 \), we calculate each subsequent term by applying the recurrence rule \( a_{n+1}=a_{n}^{2}-1 \). By consistently plugging in the previously determined term into this rule, we can find \( a_2 \) from \( a_1 \) and \( a_3 \) from \( a_2 \) and so on.

It is helpful for students to write out each step explicitly, ensuring a clear understanding of the process. By doing so, they can avoid common mistakes such as misapplying the recurrence relation or overlooking the importance of preserving the exact order of operations.

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

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{6^{n}+3^{n}}{6^{n}+n^{100}}$$

The fractal called the snowflake island (or Koch island ) is constructed as follows: Let \(I_{0}\) be an equilateral triangle with sides of length \(1 .\) The figure \(I_{1}\) is obtained by replacing the middle third of each side of \(I_{0}\) with a new outward equilateral triangle with sides of length \(1 / 3\) (see figure). The process is repeated where \(I_{n+1}\) is obtained by replacing the middle third of each side of \(I_{n}\) with a new outward equilateral triangle with sides of length \(1 / 3^{n+1}\). The limiting figure as \(n \rightarrow \infty\) is called the snowflake island. a. Let \(L_{n}\) be the perimeter of \(I_{n} .\) Show that \(\lim _{n \rightarrow \infty} L_{n}=\infty\) b. Let \(A_{n}\) be the area of \(I_{n} .\) Find \(\lim _{n \rightarrow \infty} A_{n} .\) It exists!

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{10}}{\ln ^{20} n}\right\\}$$

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. $$a_{n+1}=\frac{1}{2}\left(a_{n}+2 / a_{n}\right) ; a_{0}=2$$

Stirling's formula Complete the following steps to find the values of \(p>0\) for which the series \(\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{p^{k} k !}\) converges. a. Use the Ratio Test to show that \(\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{p^{k} k !}\) converges for \(p>2\). b. Use Stirling's formula, \(k !=\sqrt{2 \pi k} k^{k} e^{-k}\) for large \(k,\) to determine whether the series converges when \(p=2\). (Hint: \(1 \cdot 3 \cdot 5 \cdots(2 k-1)=\frac{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdots(2 k-1) 2 k}{2 \cdot 4 \cdot 6 \cdots 2 k}\) (See the Guided Project Stirling's formula and \(n\) ? for more on this topic.)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Pure 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.