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Chapter 3: Problem 24

For the sequence v defined by \(v_{n}=n !+2, \quad n \geq 1\). Find \(v_{4}\).

Short Answer

Expert verified
\(v_{4}\) equals 26.

Step by step solution

01

Understand the sequence notation

In this sequence, \(v_{n}\) is defined as the sum of the factorial of 'n' and 2. The value of 'n' changes based on the term of the sequence we are interested in. So for \(v_{4}\), 'n' will be 4.
02

Apply the factorial operation

The factorial operation '!'' is performed by multiplying all positive integers up to 'n'. So in this case, we calculate 4 factorial (written as 4!) which is 4*3*2*1 = 24.
03

Calculate \(v_{4}\)

Now using the definition of \(v_{n}\), we find that \(v_{4}\) is the sum of 4! and 2. So \(v_{4}\) = 4! + 2 = 24 + 2 = 26.

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Key Concepts

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

Sequences in Mathematics
A sequence is an ordered list of numbers, where each number is called a term. In mathematics, sequences are defined by a specific rule or formula. This means that each term in the sequence follows a particular pattern. Sequences can be finite or infinite, depending on whether there is a last term or not.
  • For example, the sequence of even numbers: 2, 4, 6, 8, ... continues indefinitely.
  • The sequence that we are working with, \(v_n = n! + 2\), is defined for each natural number \(n\).

By understanding the rule that defines a sequence, you can find any term you need. In this exercise, \(n = 4\) was substituted into the formula, allowing us to calculate the specific term of the sequence using factorial and additional operations.
Understanding the Factorial Operation
The factorial operation is denoted by an exclamation mark (!). It is a mathematical operation applied to a positive integer, \(n\), which involves multiplying all the integers from 1 to \(n\).
  • The notation \(n!\), pronounced as 'n factorial', means \(n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1\).
  • For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\).
  • The factorial of 0 is defined to be 1, which is a special case.

Factorials grow very quickly because each step involves multiplying by a larger number. This makes them important in many areas of mathematics, including combinations, permutations, and sequence definitions, such as the one in this exercise.
The Role of Mathematical Notation
Mathematical notation is like a language that helps us communicate complex ideas clearly and efficiently. It allows mathematicians to express equations, functions, and operations succinctly.
  • In sequences, notation like \(v_n\) defines how each term is calculated.
  • The factorial symbol, '!', is a compact way to express the product of a series of descending natural numbers.
  • Using notation such as \(v_n = n! + 2\), we can easily substitute values and perform calculations.

Proper understanding of mathematical notation is essential for solving problems effectively. It gives us a precise way to describe the steps involved in reaching a solution, such as computing \(v_4\) in the exercise. By consistently using standard notation, we can break down and approach complex mathematical tasks more systematically.

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Most popular questions from this chapter

For the sequence z defined by $$z_{n}=(2+n) 3^{n}, \quad n \geq 0$$. Find \(z_{2}\)

Using the sequences \(y\) and \(z\) defined by $$y_{n}=2^{n}-1, \quad z_{n}=n(n-1)$$. Find \(\left(\sum_{i=3}^{4} y_{i}\right)\left(\prod_{i=2}^{4} z_{i}\right)\)

If \(X\) and \(Y\) are sets, we define \(X\) to be equivalent to \(Y\) if there is a one-to-one, onto function from \(X\) to \(Y .\) Show that the sets \(\\{1,2, \ldots\\}\) and \(\\{2,4, \ldots\\}\) are equivalent.

Let \(X=\\{a, b\\} .\) A palindrome over \(X\) is a string \(\alpha\) for which \(\alpha=\alpha^{R}\) (i.e., a string that reads the same forward and backward). An example of a palindrome over \(X\) is bbaabb. Define a function from \(X^{*}\) to the set of palindromes over \(X\) as \(f(\alpha)=\alpha \alpha^{R} .\) Is \(f\) one-to-one? Is \(f\) onto? Prove your answers.

For the sequence z defined by $$z_{n}=(2+n) 3^{n}, \quad n \geq 0$$. Find a formula for \(z_{i}\).
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