/*! 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 18 Write the first five terms of ea... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 18

Write the first five terms of each sequence. $$ a_{n}=n+\frac{4}{n} $$

Short Answer

Expert verified
The first five terms are 5, 4, 3+\frac{4}{3}, 5, and 5+\frac{4}{5}.

Step by step solution

01

Calculate the First Term (=1)

To find the first term, substitute =1 into the formula: \(a_1 = 1+\frac{4}{1}=1+4=5\).
02

Calculate the Second Term (=2)

To find the second term, substitute =2 into the formula: \(a_2 = 2+\frac{4}{2}=2+2=4\).
03

Calculate the Third Term (=3)

To find the third term, substitute =3 into the formula: \(a_3 = 3+\frac{4}{3}=3+\frac{4}{3}=3+\frac{4}{3}\).
04

Calculate the Fourth Term (=4)

To find the fourth term, substitute =4 into the formula: \(a_4 = 4+\frac{4}{4}=4+1=5\).
05

Calculate the Fifth Term (=5)

To find the fifth term, substitute =5 into the formula: \(a_5 = 5+\frac{4}{5}=5+\frac{4}{5}\).

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.

Algebraic Sequences
Algebraic sequences are a type of mathematical sequence defined by specific formulas. Each term in the sequence relates to its position in a way defined by the formula. These sequences can be linear, quadratic, or more complex. Understanding algebraic sequences helps in predicting and calculating terms efficiently.
For instance, the given sequence is defined by the formula: \( a_{n} = n + \frac{4}{n} \).
Here, each term combines the position number \( n \) and another component derived from \( n \). Understanding this setup is key to calculating the terms.
Term Calculation
Term calculation refers to finding specific entries in a sequence using the given formula.
To calculate a term, you need to substitute the position number \( n \) into the general formula.
For example, in the sequence \(a_n = n + \frac{4}{n}\), to find the first term, we substitute \( n = 1 \):
\( a_1 = 1 + \frac{4}{1} = 1 + 4 = 5 \).
You follow this process for other terms by changing the value of \( n \) accordingly.
Mathematical Formulas
Mathematical formulas are used to define sequences and other relationships. In the context of sequences, formulas allow quick calculation of any term in the sequence.
For the sequence \( a_n = n + \frac{4}{n} \), the formula tells us how to combine the term number \( n \) and another part that depends on \( n \).
Formulas help to understand the nature of a sequence, whether it's increasing, decreasing, or following some other pattern.
Substitution Method
The substitution method involves replacing a variable in a formula with an actual value. This is a straightforward process and is essential in calculating terms in sequences.
For our sequence \( a_{n} = n + \frac{4}{n} \), to find the second term, we substitute \( n = 2 \):
\( a_2 = 2 + \frac{4}{2} = 2 + 2 = 4 \).
By systematically substituting values of \( n \), we can determine any term in the sequence.
This method is simple yet powerful for evaluating mathematical expressions and sequences.

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

Solve each problem. The number of mutual funds operating in the United States each year during the period 2012 through 2016 is given in the table. To the nearest whole number, what was the average number of mutual funds operating per year during the given period? $$ \begin{array}{|c|c|} \hline \text { Year } & \text { Number of Mutual Funds } \\ 2012 & 8744 \\ 2013 & 8972 \\ 2014 & 9258 \\ 2015 & 9517 \\ 2016 & 9511 \\ \hline \end{array} $$

Write each series as a sum of terms and then find the sum. $$ \sum_{i=1}^{6}(-1)^{i} \cdot 2 $$

When dropped from a certain height, a ball rebounds to \(\frac{3}{5}\) of the original height. How high will the ball rebound after the fourth bounce if it was dropped from a height of \(10 \mathrm{ft}\) ? Round to the nearest tenth.

Find the indicated term for each geometric sequence $$ \frac{2}{3}, \frac{1}{3}, \frac{1}{6}, \ldots ; \quad a_{18} $$

Use mathematical induction to prove that each statement is true for every positive integer value of \(n.\) $$\frac{1}{1 \cdot 4}+\frac{1}{4 \cdot 7}+\frac{1}{7 \cdot 10}+\cdots+\frac{1}{(3 n-2)(3 n+1)}=\frac{n}{3 n+1}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Decision 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.