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Chapter 9: Problem 43

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n !}{n^{2}} $$

Short Answer

Expert verified
The first four terms are 1, \( \frac{1}{2} \), \( \frac{2}{3} \), \( \frac{3}{2} \).

Step by step solution

01

Evaluate the formula for n=1

Substitute 1 into the formula: \( a_{1} = \frac{1!}{1^2} = \frac{1}{1} = 1 \). Therefore, the first term is 1.
02

Evaluate the formula for n=2

Substitute 2 into the formula: \( a_{2} = \frac{2!}{2^2} = \frac{2 \times 1}{4} = \frac{2}{4} = \frac{1}{2} \). Therefore, the second term is \( \frac{1}{2} \).
03

Evaluate the formula for n=3

Substitute 3 into the formula: \( a_{3} = \frac{3!}{3^2} = \frac{3 \times 2 \times 1}{9} = \frac{6}{9} = \frac{2}{3} \). Therefore, the third term is \( \frac{2}{3} \).
04

Evaluate the formula for n=4

Substitute 4 into the formula: \( a_{4} = \frac{4!}{4^2} = \frac{4 \times 3 \times 2 \times 1}{16} = \frac{24}{16} = \frac{3}{2} \). Therefore, the fourth term is \( \frac{3}{2} \).

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Key Concepts

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

Factorial
The concept of a factorial is exciting and unique in mathematics. It is denoted by an exclamation mark (!). The factorial of a non-negative integer, say \( n \), is the product of all positive integers less than or equal to \( n \). So, for instance:
  • \( 0! = 1 \) by definition.
  • \( 1! = 1 \)
  • \( 2! = 2 \times 1 = 2 \)
  • \( 3! = 3 \times 2 \times 1 = 6 \)
  • \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
Factorials grow very quickly with larger numbers. They are useful in many areas, such as permutations and combinations, describing sequences, and more. Understanding factorials helps in evaluating expressions like the one used in your exercise.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. In your exercise, the algebraic expression is \( \frac{n!}{n^{2}} \). It combines factorials and exponents.
  • The numerator, \( n! \), represents the factorial.
  • The denominator, \( n^{2} \), represents the square of \( n \).
By substituting values into this expression, you can evaluate it to find specific terms of a sequence. Understanding how to manipulate algebraic expressions is crucial to solving math problems efficiently.
Step-by-Step Problem Solving
Solving math problems step-by-step is a powerful method to tackle seemingly complex problems. It allows you to break down a problem into manageable parts. In this exercise, each step represents evaluating the expression for different values of \( n \):
  • Start with \( n = 1 \), evaluate the expression, and record the result.
  • Repeat the process for \( n = 2, 3, \) and \( 4 \).
This systematic approach ensures that nothing is overlooked, and each computation builds on the previous one. It's a technique that can be applied across various mathematical disciplines and problem types, enhancing both understanding and precision in problem-solving.
Evaluating Sequences
Evaluating sequences involves finding specific terms within a sequence as defined by a formula. In your exercise, the formula \( a_{n} = \frac{n!}{n^{2}} \) was given to write out the first four terms.
  • By substituting \( n = 1 \), the first term \( a_1 \) was found as 1.
  • For \( n = 2 \), the second term \( a_2 \) is \( \frac{1}{2} \).
  • With \( n = 3 \), the third term \( a_3 \) is \( \frac{2}{3} \).
  • Finally, substituting \( n = 4 \) yields the fourth term \( a_4 \) as \( \frac{3}{2} \).
Each calculation provides a snapshot of the sequence, illustrating how each term relates to \( n \). Evaluating sequences is an important skill in mathematics, as it connects formulas to tangible values and helps students explore patterns and relationships within numbers.

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Most popular questions from this chapter

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum less than 6 or greater than \(9 .\)

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting 20 numbers from the numbers 1 to 80 . After the player makes his selections, 20 winning numbers are randomly selected from numbers 1 to 80 . A win occurs if the player has correctly selected 3,4 , or 5 of the 20 winning numbers. (Round all answers to the nearest hundredth of a percent.) What is the percent chance that a player selects exactly 3 winning numbers?

Use the geometric series \(\sum_{k=1}^{\infty}\left(\frac{1}{2}\right)^{k}\). What number does \(S_{n}\) seem to be approaching in the graph? Find the sum to explain why this makes sense.

Use this data for the exercises that follow: In 2013 , there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over). \(^{[34]}\) If you meet a U.S. citizen, what is the percent chance that the person is elderly? (Round to the nearest tenth of a percent.)

For the following exercises, four coins are tossed. Find the probability of tossing exactly two heads or at least two tails.
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