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Chapter 13: Problem 19

List the first five terms of each sequence. \(\left\\{c_{n}\right\\}=\left\\{(-1)^{n+1} n^{2}\right\\}\)

Short Answer

Expert verified
1, -4, 9, -16, 25

Step by step solution

01

Understand the General Formula

The given sequence is defined by the formula \(c_n = (-1)^{n+1} n^2\). This means each term in the sequence is determined by plugging in the values of \(n\) starting from 1.
02

Calculate the First Term

For \(n = 1\): \[ c_1 = (-1)^{1+1} \times 1^2 = (-1)^2 \times 1 = 1 \]
03

Calculate the Second Term

For \(n = 2\): \[ c_2 = (-1)^{2+1} \times 2^2 = (-1)^3 \times 4 = -4 \]
04

Calculate the Third Term

For \(n = 3\): \[ c_3 = (-1)^{3+1} \times 3^2 = (-1)^4 \times 9 = 9 \]
05

Calculate the Fourth Term

For \(n = 4\): \[ c_4 = (-1)^{4+1} \times 4^2 = (-1)^5 \times 16 = -16 \]
06

Calculate the Fifth Term

For \(n = 5\): \[ c_5 = (-1)^{5+1} \times 5^2 = (-1)^6 \times 25 = 25 \]

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Key Concepts

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

general formula
The general formula given for the sequence is denoted by \( c_n = (-1)^{n+1} n^2 \). This formula helps determine each term in the sequence by substituting different values of \ \ into the equation.
In simpler terms, \( (-1)^{n+1} \) alternates the sign (positive or negative) of each term and \( n^2 \) squares the term number.
This alternation in sign is crucial as it creates a unique pattern within the sequence.
We will use this general formula to find each term step-by-step.
alternating sequence
An alternating sequence is a sequence in which the terms change signs as they progress.
In this sequence, \( (-1)^{n+1} \) is what causes the alternation.
Let's break it down:
  • When \( n \) is odd (1, 3, 5,...), \( (-1)^{n+1} \) becomes even, which results in \( (-1)^{even} = 1 \).
  • When \( n \) is even (2, 4, 6,...), \( (-1)^{n+1} \) becomes odd, which results in \( (-1)^{odd} = -1 \).
Thus, odd \( n \) gives positive and even \( n \) gives negative results.
This mechanism causes the sequence to flip signs alternatively with each term.
term calculation
To find the sequence terms, we follow these steps:
- Plug the value of \( n \) starting from 1 into the formula.
- Compute the value step-by-step.
Let's calculate the first five terms:
  • For \( n = 1 \), \( c_1 = (-1)^{1+1} \times 1^2 = (-1)^2 \times 1 = 1 \).
  • For \( n = 2 \), \( c_2 = (-1)^{2+1} \times 2^2 = (-1)^3 \times 4 = -4 \).
  • For \( n = 3 \), \( c_3 = (-1)^{3+1} \times 3^2 = (-1)^4 \times 9 = 9 \).
  • For \( n = 4 \), \( c_4 = (-1)^{4+1} \times 4^2 = (-1)^5 \times 16 = -16 \).
  • For \( n = 5 \), \( c_5 = (-1)^{5+1} \times 5^2 = (-1)^6 \times 25 = 25 \).
Each calculation follows the same basic steps but alternates the sign and squares the number accordingly.
mathematical sequences
A mathematical sequence is a list of numbers that follows a specific pattern.
Sequences can be found everywhere in mathematics and form the basis for many complex theories.
In this exercise, our sequence is both arithmetic (because of the squaring) and geometric (due to the alternating negative and positive signs).
Understanding the pattern in a sequence helps simplify its representation and further calculations.
exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent.
In this sequence, the term number \( n \) is the base, and it is squared, meaning the exponent is 2 (n^2).
Specifically, squaring a number means multiplying it by itself:
  • \( 1^2 = 1 \)
  • \( 2^2 = 4 \)
  • \( 3^2 = 9 \)
  • \( 4^2 = 16 \)
  • \( 5^2 = 25 \)
This exponentiation creates the magnitude of each term, while the \( (-1)^{n+1} \) factor alternates its sign.
Combining these two operations results in the unique sequence presented.

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Most popular questions from this chapter

Challenge Problem Paper Creases If a sheet of paper is folded in half by folding the top edge down to the bottom edge, one crease will result. If the folded paper is folded in the same manner, the result is three creases. With each fold, the number of creases can be defined recursively by \(c_{1}=1, c_{n+1}=2 c_{n}+1\) (a) Find the number of creases for \(n=3\) and \(n=4\) folds. (b) Use the given information and your results from part (a) to find a formula for the number of creases after \(n\) folds, \(c_{n}\), in terms of the number of folds alone. (c) Use the Principle of Mathematical Induction to prove that the formula found in part (b) is correct for all natural numbers. (d) Tosa Tengujo is reportedly the world's thinnest paper with a thickness of \(0.02 \mathrm{~mm}\). If a piece of this paper could be folded 25 times, how tall would the stack be?

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