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Chapter 9: Problem 74

In Exercises 71-76, write the first five terms of the sequence. (Assume that \( n \) begins with 0.) \( a_n = \dfrac{n^2}{(n + 1)!} \)

Short Answer

Expert verified
The first five terms of the sequence are: 0, 0.5, 0.66, 0.375, 0.133

Step by step solution

01

Substitute n=0 into sequence equation

When n=0, substitute 0 into the equation \( a_n = \dfrac{n^2}{(n + 1)!} \), we get \( a_0 = \dfrac{0^2}{(0 + 1)!}=\frac{0}{1}=0 \)
02

Substitute n=1 into sequence equation

When n=1, substitute 1 into the equation \( a_n = \dfrac{n^2}{(n + 1)!} \), we get \( a_1 = \dfrac{1^2}{(1 + 1)!}=\frac{1}{2}=0.5 \)
03

Substitute n=2 into sequence equation

When n=2, substitute 2 into the equation \( a_n = \dfrac{n^2}{(n + 1)!} \), we get \( a_2 = \dfrac{2^2}{(2 + 1)!}=\frac{4}{6}=0.\overline{66} \)
04

Substitute n=3 into sequence equation

When n=3, substitute 3 into the equation \( a_n = \dfrac{n^2}{(n + 1)!} \)(where ! represents factorial operation), we get \( a_3 = \dfrac{3^2}{(3 + 1)!}=\frac{9}{24}=0.375 \)
05

Substitute n=4 into sequence equation

When n=4, substitute 4 into the equation \( a_n = \dfrac{n^2}{(n + 1)!} \), we get \( a_4 = \dfrac{4^2}{(4 + 1)!}=\frac{16}{120}=0.133\overline{3} \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Factorial
The concept of a factorial is essential in understanding sequences like the one given in the exercise. A factorial, denoted by the symbol \(!\), is a product of all positive integers up to a certain number. For instance, the factorial of 3, written as \(3!\), is calculated as \(3 \times 2 \times 1 = 6\). For every integer \(n\),
  • \(0! = 1\) by definition
  • \(1! = 1\)
  • \(2! = 2 \times 1 = 2\)
  • \(3! = 3 \times 2 \times 1 = 6\)
  • and so forth.
Factorials grow very fast as numbers increase, meaning even small numbers produce large factorial values quickly. This is an important aspect when calculating terms in sequences, as it might affect whether sequence terms are large or small.
Sequence Term Calculation for a Given Formula
Calculating each term of a sequence given a specific formula involves substituting values into the sequence's formula. The sequence we are examining is given by the expression:\[ a_n = \dfrac{n^2}{(n + 1)!} \]To find each term:
  • Start with \(n = 0\) and substitute it into the formula.
  • Continue replacing \(n\) with each subsequent integer until you have the desired number of terms.
For example, substituting \(n = 0\) gives \(a_0 = \dfrac{0^2}{(0 + 1)!} = 0\).
Substituting \(n = 1\) gives \(a_1 = \dfrac{1^2}{(1 + 1)!} = 0.5\), and so on.
Each substitution provides a clear view of individual terms in the sequence. It shows how changing \(n\) affects the outcome significantly, given the factorial growth in the denominator.
Exploring Mathematical Sequences
A mathematical sequence is essentially a list of numbers ordered in a specific way according to a given rule or formula. This order can be crucial for discovering the nature and properties of the sequence. In this exercise, we are looking into a sequence defined by both a polynomial expression and a factorial in its denominator:
This specific kind of sequence
  • shows a unique pattern as \(n\) increases.
  • Factors like factorial in the denominator contribute to how fast the values decrease.
In general, sequences can come in various forms:
  • Arithmetic (where there's a constant difference between terms)
  • Geometric (where each term is a constant multiple of the previous)
  • Other special forms like Fibonacci or alternating sequences.
Understanding the general setup of sequences helps in recognizing how they may behave, whether they converge or diverge, and their importance in broader mathematical contexts.

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Most popular questions from this chapter

In Exercises 31 - 34, find the probability for the experiment of drawing two marbles (without replacement) from a bag containing one green, two yellow, and three red marbles. The marbles are of different colors.

In Exercises 21 - 24, find the probability for the experiment of selecting one card from a standard deck of \( 52 \) playing cards. The card is a face card.

In Exercises 9 - 14, determine the sample space for the experiment. A sales representative makes presentations about a product in three homes per day. In each home, there may be a sale (denote by \( S \)) or there may be no sale(denote by \( F \)).

Write a short paragraph defining the following. (a) Sample space of an experiment (b) Event (c) The probability of an event \( E \) in a sample space \( S \) (d) The probability of the complement of \( E \)

In Exercises 9 - 14, determine the sample space for the experiment. A taste tester has to rank three varieties of yogurt, \( A \), \( B \),and \( C \), according to preference.
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