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Chapter 13: Problem 6

Compute the first four terms in each sequence. For Exercises \(1-10,\) assume that each sequence is defined for \(n \geq 1 ;\) in Exercises \(11-14,\) assume that each sequence is defined for \(n \geq 0\) $$c_{n}=(-1)^{n}\left(2^{n}\right)$$

Short Answer

Expert verified
The first four terms are -2, 4, -8, 16.

Step by step solution

01

Identify the Formula

The formula for the sequence is given as \( c_n = (-1)^n (2^n) \). This sequence alternates signs based on \( (-1)^n \) and grows in magnitude as \( 2^n \).
02

Calculate the First Term

To find the first term \( c_1 \), substitute \( n = 1 \) into the formula: \[ c_1 = (-1)^1 (2^1) = -2 \].
03

Calculate the Second Term

To find the second term \( c_2 \), substitute \( n = 2 \) into the formula: \[ c_2 = (-1)^2 (2^2) = 4 \].
04

Calculate the Third Term

To find the third term \( c_3 \), substitute \( n = 3 \) into the formula: \[ c_3 = (-1)^3 (2^3) = -8 \].
05

Calculate the Fourth Term

To find the fourth term \( c_4 \), substitute \( n = 4 \) into the formula: \[ c_4 = (-1)^4 (2^4) = 16 \].
06

List the First Four Terms

Now we list the first four terms of the sequence: \( -2, 4, -8, 16 \). This sequence alternates signs and doubles in magnitude.

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Key Concepts

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

Alternating Sequences
In mathematics, an alternating sequence is a pattern where the signs of the terms switch back and forth. For instance, positive numbers may alternate with negative ones, or vice versa.
Alternating sequences are characterized by terms like (-1)^n, which ensures that the sign changes with each successive term.
This creates a sequence that can look something like this:
  • First term positive
  • Second term negative
  • Third term positive
  • Fourth term negative
This variation in sign can be crucial for representing oscillating behaviors in various mathematical contexts. Alternating sequences can often be used to model phenomena in physics and engineering, where systems naturally alternate between states.
Exponential Growth
Exponential growth describes a process where the rate of growth is proportional to the current size, resulting in a rapid increase over time.
In sequences, we see this type of growth when terms involve powers of a constant, such as 2^n. This is important because it shows how quickly values can escalate in terms of their magnitude.
For example, if you start with 1 and repeatedly multiply by 2, you'll observe:
  • 1 becomes 2
  • 2 becomes 4
  • 4 becomes 8
  • 8 becomes 16
The pattern is clear; each term is twice the size of its predecessor. Understanding exponential growth helps in fields like finance, biology, and computer science, where understanding rapid increases is essential.
Sequence Formula
A sequence formula provides a way to generate the terms in a sequence. In mathematical exercises, being able to understand and manipulate these formulas is key to solving problems.
For the formula given in the original exercise, \( c_n = (-1)^n \cdot (2^n) \), two components are combined:
  • \((-1)^n\) - ensures that terms alternate in sign
  • \(2^n\) - causes the magnitude to double with each increasing term
By substituting different values of \(n\) into this formula, each term in the sequence can be calculated easily.
This foundational understanding is vital in tackling more complex sequences and series, where sequence expressions can model various real-world processes.
Term Calculation
Term calculation in sequences requires substituting index values into the sequence formula efficiently. For instance, in the given formula \(c_n = (-1)^n \cdot (2^n)\), each term can be calculated as follows:
  • For the first term, \(n=1\), we find \(c_1 = (-1)^1 \times 2^1 = -2\).
  • For the second term, \(n=2\), we calculate \(c_2 = (-1)^2 \times 2^2 = 4\).
  • For the third term, \(n=3\), substitution gives us \(c_3 = (-1)^3 \times 2^3 = -8\).
  • For the fourth term, \(n=4\), it results in \(c_4 = (-1)^4 \times 2^4 = 16\).
Calculating terms this way is systematic; once the pattern is known, executing it for each value of \(n\) is straightforward. Term calculation is a basic yet crucial skill in mathematics, serving as a bridge to understanding more complex mathematical concepts and sequences.

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Most popular questions from this chapter

Find the coefficient of the term containing \(a^{4}\) in the expansion of \((\sqrt{a}-\sqrt{x})^{10}\).

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Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$ \left(\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right) \div\left(\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right) $$

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an arithmetic sequence, and let \(S_{k}\) denote the sum of the first \(k\) terms. If \(S_{n} / S_{m}=n^{2} / m^{2}\) show that $$\frac{a_{n}}{a_{m}}=\frac{2 n-1}{2 m-1}$$

Find the coefficient of the term containing \(a^{8}\) in the expansion of \([a-(2 / \sqrt{a})]^{14}\).
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