Problem 69 - Algebra for College Students

Question

For what natural numbers \(n\) does \((-1)^{n}=-1\) ? For what natural numbers \(n\) does \((-1)^{n}=1\) ? Explain your answers.

Step-by-Step Solution

Answer

\(n\) is odd for \((-1)^n = -1\) and even for \((-1)^n = 1\).

Step 1: Understanding the problem

We need to find the natural numbers \(n\) such that \((-1)^n = -1\), and those such that \((-1)^n = 1\). Natural numbers are positive integers: 1, 2, 3, and so on.

Step 2: Evaluating the expression \((-1)^n\)

The expression \((-1)^n\) depends on whether \(n\) is odd or even. If \(n\) is odd, \((-1)^n = -1\), and if \(n\) is even, \((-1)^n = 1\).

Step 3: Identifying odd \(n\) for \((-1)^n = -1\)

When \(n\) is an odd natural number, \((-1)^n = -1\). Examples of odd natural numbers include 1, 3, 5, etc. Thus, \(n\) must be an odd number for \((-1)^n = -1\).

Step 4: Identifying even \(n\) for \((-1)^n = 1\)

When \(n\) is an even natural number, \((-1)^n = 1\). Examples of even natural numbers include 2, 4, 6, etc. Therefore, \(n\) must be an even number for \((-1)^n = 1\).

Step 5: Writing the solution

For the condition \((-1)^n = -1\), \(n\) must be an odd natural number. For \((-1)^n = 1\), \(n\) must be an even natural number.

Key Concepts

Even Numbers

Even numbers are natural numbers that can be divided by 2 without leaving a remainder. They are like members of a club that follow a simple rule — they pair up perfectly. Examples of even numbers include:

• 2
• 4
• 6
• 8
• 10

The defining characteristic of even numbers is their ability to be expressed as 2 times another natural number. This can be written mathematically as:\[ n = 2k \]where \( n \) is an even number, and \( k \) is a natural number. In the context of the problem where we evaluate \((-1)^n = 1\), we identify that when \( n \) is even, the expression yields 1. This results from the fact that multiplying negative one by itself an even number of times cancels out the negative sign, leading to a positive 1. You can visualize it this way:- Two negatives make a positive. Think of two negatives like flipping a coin twice from negative to positive, leading to a positive outcome.

Odd Numbers

Odd numbers, on the other hand, are those natural numbers that do not evenly divide by 2. Whenever you try to make pairs with odd numbers, there’s always one number left without a partner. Examples include:

• 1
• 3
• 5
• 7
• 9

These numbers are expressed mathematically as:\[ n = 2k + 1 \]where \( n \) is an odd number and \( k \) is a natural number. In the problem, for \((-1)^n = -1\), we determine that when \( n \) is odd, the expression is equal to \(-1\). This is because an odd number of negative multiplications results in a negative product. When you multiply negative one an odd number of times, the end result remains negative. - Think of it as flipping a light switch an odd number of times. If the switch started in the "off" (negative) position and you flipped it an odd number of times, it will end up "off" again.

Exponents

Exponents are a shorthand way to describe repeated multiplication of a number by itself. In other words, an exponent tells you how many times to multiply the base number. Consider the expression \((a)^b\).Here:

• \(a\) is the base
• \(b\) is the exponent

The expression tells us to multiply \(a\) by itself \(b\) times. For example, \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).Understanding exponents is key when solving the problem with \((-1)^n\). The base here is -1, and the exponent \(n\) determines how many times -1 is multiplied. The resulting effect is based on whether \(n\) is even or odd:- If \(n\) is even, the negative signs pair up to form positive results.- If \(n\) is odd, one negative is always left unpaired, resulting in a negative product.This concept illustrates why calculators and mental math alike follow specific patterns when evaluating powers of negative numbers. Exponents simplify complex multiplication and help to predict outcomes based on the parity (even or odd nature) of the number involved.