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Chapter 11: Problem 6

List the first five terms of the sequence. $$ \\{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2 n)\\} $$

Short Answer

Expert verified
The first five terms are 2, 8, 48, 384, and 3840.

Step by step solution

01

Understand the Sequence

The given sequence \( \{2 \cdot 4 \cdot 6 \cdot \cdots \cdot(2n)\} \) is the product of the first \( n \) even numbers. To list the first five terms, we need to compute the product starting from \( n = 1 \) and continue up to \( n = 5 \).
02

Compute the First Term

For \( n = 1 \), the term is \( 2 \cdot 1 = 2 \). There is only one factor in this product.
03

Compute the Second Term

For \( n = 2 \), the term is \( 2 \cdot 4 = 8 \). We multiply the first two even numbers.
04

Compute the Third Term

For \( n = 3 \), the term is \( 2 \cdot 4 \cdot 6 = 48 \). We multiply the first three even numbers together.
05

Compute the Fourth Term

For \( n = 4 \), the term is \( 2 \cdot 4 \cdot 6 \cdot 8 = 384 \). This is the product of the first four even numbers.
06

Compute the Fifth Term

For \( n = 5 \), the term is \( 2 \cdot 4 \cdot 6 \cdot 8 \cdot 10 = 3840 \). We continue to multiply the first five even numbers.

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Key Concepts

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

Understanding the Product of Even Numbers
A sequence that involves the product of even numbers is an arrangement of numbers where each term is the result of multiplying consecutive even numbers. Even numbers are numbers like 2, 4, 6, and so on, each divisible by 2. When we talk about the product of even numbers, we mean multiplying these even numbers together. For example, if you multiply 2, 4, and 6, you obtain a product.

This kind of product is significant in understanding growth as each additional term involves more multiplication, rapidly increasing the product size. The sequence in our exercise involves multiplying the first n even numbers, providing terms that are progressively larger as n increases.
Exploring Sequence Terms
Sequence terms refer to the individual elements in a sequence, each formed according to a specific rule or pattern. In our context, each term is a result of multiplying a series of even numbers. When we list the sequence terms, we begin with the simplest form for low values of n.

- For n=1, the first term is simply 2. - For n=2, the term is 2 multiplied by the next even number, 4. - As n increases, more even numbers are multiplied together to form subsequent terms. Understanding sequence terms in a product of even numbers sequence helps grasp the concept of sequences which build on themselves by rules and patterns.
Calculation of Sequences
Calculating a sequence involves determining each term based on an established rule or formula. In our example, we calculate terms by progressively multiplying even numbers. To start, identify the formula or rule鈥攈ere, we multiply even numbers up to 2n.

The steps to calculate include: - Start with the first term, using the smallest number in the sequence. - Multiply each consecutive number by the next even number to form subsequent terms, up to the desired n value. This systematic approach to calculation ensures that each sequence term is derived correctly, illustrating how sequences expand exponentially as n increases.
Introduction to Mathematical Series
A mathematical series is essentially the sum of terms in a sequence, but it first requires understanding how sequences themselves work. In our sequence, we focus on the product of even numbers rather than summing them, but the foundational understanding of sequences and series is interconnected.

For instance, the sequence 2, 8, 48, 384, and 3840 shows the results of multiplying even numbers up to specific points, relying on sequences to build toward forming a potential series should their sums be considered. Sequences and series are both fundamental concepts in math that help with understanding relationships between numbers.
Deep Dive into Even Number Multiplication
Multiplying even numbers involves consecutive even numerals such as 2, 4, 6, and so forth. This multiplication method is powerful, as each step reveals how rapid the growth is with each additional multiplier.

Steps for even number multiplication: - Identify the sequence of even numbers you want to multiply. - Multiply each number in the sequence step-by-step. - Assess the result to understand how multiplication affects growth, especially useful in sequences. Multiplying even numbers, like in our sequence example, shows how quickly values can grow and illustrates exponential increases. This is a crucial insight for anyone studying products and sequences.

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Most popular questions from this chapter

The resistivity \(\rho\) of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters \((\Omega-m) .\) The resistivity of a given metal depends on the temperature according to the equation $$\rho(t)=\rho_{20} e^{\alpha(t-20)}$$ where \(t\) is the temperature in \(^{\circ} \mathrm{C}\) . where \(t\) is the temperature in \(^{\circ} \mathrm{C} .\) There are tables that list the values of \(\alpha\) (called the temperature coefficient) and \(\rho_{20}\) (the resistivity at \(20^{\circ} \mathrm{C} )\) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for \(\rho(t)\) by its first- or second-degree Taylor polynomial at \(t=20\) . (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give \(\alpha=0.0039 /^{\circ} \mathrm{C}\) and \(\rho_{20}=1.7 \times 10^{-8} \Omega-\mathrm{m} .\) Graph the resistivity of copper and the linear and quadratic approximations for \(-250^{\circ} \mathrm{C} \leqslant t \leqslant 1000^{\circ} \mathrm{C}\) (c) For what values of \(t\) does the linear approximation agree with the exponential expression to within one percent?

\(f(x)=\sin ^{2} x\) \([\)Hint\(:\) Use \(\sin ^{2} x=\frac{1}{2}(1-\cos 2 x) . \)]

Use multiplication or division of power series to find the first three nonzero terms in the Maclaurin series for the function. \(y=\sec x\)

(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number a. (b) Use Taylor's Inequality to estimate the accuracy of the approximation \(f(x) \approx T_{n}(x)\) when \(x\) lies in the given interval. (c) Check your result in part (b) by graphing \(\left|R_{n}(x)\right|\) $$f(x)=\ln (1+2 x), \quad a=1, \quad n=3, \quad 0.5 \leqslant x \leqslant 1.5$$

Find the Maclaurin series of \(f\) (by any method) and its radius of convergence. Graph \(f\) and its first few Taylor polynomials on the same screen. What do you notice about the relationship between these polynomials and \(f\) ? \(f(x)=e^{-x^{2}}+\cos x\)
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