/*! 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 10 List the first five terms of the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 10

List the first five terms of the sequence. $$ a_{1}=6, \quad a_{n+1}=\frac{a_{n}}{n} $$

Short Answer

Expert verified
The first five terms are 6, 6, 3, 1, and \( \frac{1}{4} \).

Step by step solution

01

Understand the Initial Term

The problem gives the initial term of the sequence as \( a_1 = 6 \). This means the first term in the sequence is 6.
02

Calculate the Second Term

Using the recursive formula \( a_{n+1} = \frac{a_n}{n} \), we find the second term: \( a_2 = \frac{a_1}{1} = \frac{6}{1} = 6 \).
03

Calculate the Third Term

To find the third term, use the formula again: \( a_3 = \frac{a_2}{2} = \frac{6}{2} = 3 \).
04

Calculate the Fourth Term

For the fourth term, apply the formula: \( a_4 = \frac{a_3}{3} = \frac{3}{3} = 1 \).
05

Calculate the Fifth Term

Finally, calculate the fifth term using the formula: \( a_5 = \frac{a_4}{4} = \frac{1}{4} \).
06

Solution

The first five terms of the sequence are: 6, 6, 3, 1, and \( \frac{1}{4} \).

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.

Understanding Recursive Formulas
Recursive formulas play a critical role in defining sequences in calculus. They describe how each term in a sequence is derived from one or more previous terms. In simpler terms, a recursive formula allows you to find the next term in a sequence if you know the preceding terms.
For instance, in the exercise given, the recursive formula is \( a_{n+1} = \frac{a_n}{n} \). This means that to find the term \( a_{n+1} \), you need to divide the previous term \( a_n \) by \( n \). Recursive formulas are helpful because they tell us exactly how to move from one term to the next, simplifying the process of finding subsequent numbers in a sequence.
Sequence Terms Explained
The terms of a sequence are the individual elements or numbers in a sequence that follow a specific pattern. In calculus, understanding these terms is crucial for solving sequence-related problems.
In our given exercise, the terms are generated using the recursive formula starting from an initial term. The initial term \( a_1 \) is given as 6. Thus:
  • The first term \( a_1 = 6 \)
  • The second term \( a_2 = \frac{a_1}{1} = 6 \)
  • The third term \( a_3 = \frac{a_2}{2} = 3 \)
  • The fourth term \( a_4 = \frac{a_3}{3} = 1 \)
  • The fifth term \( a_5 = \frac{a_4}{4} = \frac{1}{4} \)
Each term relies on the previous one, following the pattern established by the recursive formula.
Approaching Calculus Problem Solving
Solving calculus problems involving sequences requires a step-by-step approach to ensure a thorough understanding.
Here鈥檚 a simple strategy on how to tackle such problems:
  • Identify the initial term: Look for the first term given, as it's the starting point of your sequence.
  • Understand the recursive formula: Recognize how each term transitions to the next. This involves understanding how you will use the current term to determine the next one.
  • Calculate each term systematically: Begin with the initial term and apply the recursive formula to find the subsequent terms. Make sure to follow this process for each term in the sequence.
  • Verify your results: Once you generate the terms using the formula, double-check by recalculating or visualizing the pattern to confirm correctness.
By following these steps, you can approach calculus problems effectively and arrive at the correct solution to sequence-related questions.

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

Suppose you know that \(f^{(x)}(4)=\frac{(-1)^{n} n !}{3^{n}(n+1)}\) and the Taylor series of \(f\) centered at 4 converges to \(f(x)\) for all \(x\) in the interval of convergence. Show that the fifth-degree Taylor polynomial approximates \(f(5)\) with error less than 0.0002 .

Find the Taylor series for \(f(x)\) centered at the given value of \(a\). [ Assume that \(f\) has a power series expansion. Do not show that \(\left.R_{n}(x) \rightarrow 0 .\right]\) Also find the associated radius of convergence. $$ f(x)=x^{6}-x^{4}+2, \quad a=-2 $$

Use series to evaluate the limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{1+x}-1-\frac{1}{2} x}{x^{2}} $$

Use a computer algebra system to find the Taylor polynomials \(T_{n}\) centered at \(a\) for \(n=2,3,4,5\). Then graph these polynomials and \(f\) on the same screen. $$ f(x)=\sqrt[3]{1+x^{2}}, \quad a=0 $$

Prove Taylor's Inequality for \(n=2,\) that is, prove that if \(\left|f^{\prime \prime}(x)\right| \leqslant M\) for \(|x-a| \leqslant d\), then $$ \left|R_{2}(x)\right| \leqslant \frac{M}{6}|x-a|^{3} \quad \text { for }|x-a| \leqslant d $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

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.