Problem 9 Show that the square of every od... [FREE SOLUTION]

Chapter 1: Problem 9

Show that the square of every odd integer is of the form \(8 m+1\).

Short Answer

Expert verified

Question: Prove that the square of every odd integer is in the form of \(8m+1\), where \(m\) is an integer. Answer: We have shown that the square of an odd integer \((2n+1)^2\) can be expressed in the form \(8m+1\), where \(m = \frac{1}{2}n^2 + \frac{1}{2}n\) and \(m\) is an integer. This completes the proof and confirms that the square of every odd integer is indeed of the form \(8m + 1\).

Step by step solution

Write the odd integer in the form of \(2n+1\)

An odd integer can be expressed as \(2n+1\), where \(n\) is an integer. This is because multiplying any integer by 2 will result in an even number, and adding 1 to an even number results in an odd number.

Calculate the square of the odd integer

Now, we will find the square of the odd integer \((2n+1)\). This is done by multiplying the odd integer by itself: \((2n+1)^2\)

Simplify the expression

Next, we will simplify the expression \((2n+1)^2\) by using the distributive property (also known as FOIL method) and combining like terms: \((2n+1)^2 = (2n+1)(2n+1) = 4n^2 + 4n + 1\)

Rewrite the expression in the form \(8m+1\)

Now, we will try to express the simplified expression in the form \(8m+1\). First, note that \(4n^2 + 4n\) is an even number because it is a multiple of 2. So we can rewrite it as \(2(2n^2+2n)\): \(4n^2 + 4n + 1 = 2(2n^2 + 2n) + 1\) Now, notice that \(2n^2 + 2n\) is a multiple of 4, since \(2n^2\) and \(2n\) are both multiples of 4. Therefore, we can rewrite the expression as: \(2(2n^2 + 2n) + 1 = 8(\frac{2n^2 + 2n}{4}) + 1 = 8(\frac{1}{2}n^2 + \frac{1}{2}n) + 1 = 8m + 1\) Here, we let \(m = \frac{1}{2}n^2 + \frac{1}{2}n\). Since \(n\) is an integer, and we're adding two integers \(\frac{1}{2}n^2\) and \(\frac{1}{2}n\), \(m\) must also be an integer.

Conclude the proof

We have shown that the square of an odd integer \((2n+1)^2\) can be expressed in the form \(8m+1\), where \(m = \frac{1}{2}n^2 + \frac{1}{2}n\) and \(m\) is an integer. This completes the proof and confirms that the square of every odd integer is indeed of the form \(8m + 1\).

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Key Concepts

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

Odd integers

Odd integers are numbers that cannot be divided evenly by 2. They are of the form \(2n+1\), where \(n\) is any integer. This form comes from the fact that multiplying any whole number by 2 gives an even number, and adding 1 to this even number makes it odd.

In simpler terms, odd numbers include familiar numbers like 1, 3, 5, 7, etc. These numbers have a difference of 2 between consecutive odd integers. Understanding the initial representation \(2n+1\) helps in various mathematical problems, like proving certain properties of odd numbers.

Key characteristics of odd numbers include:

They end in 1, 3, 5, 7, or 9 in the decimal system.
The sum or difference of two odd numbers is even.
The product of two odd numbers is always odd.

Mathematical proof

A mathematical proof is a logical argument that confirms the truth of a given statement. In mathematics, proofs are essential to establish facts with certainty. Every proof begins with known facts, often called axioms, and progresses through a logical sequence of steps to arrive at a conclusion.

The proof format used in the exercise is called a "direct proof," which assumes an odd integer in the form \(2n+1\) and logically derives the form of its square as \(8m+1\). This requires breaking down the expression, simplifying it, and then reorganizing it to match the form intended.

Key components of clear and helpful proofs include:

Starting with clear definitions of the terms and numbers involved.
Using logical steps that follow from known principles or previous results.
Communicating each step clearly to show how the conclusions are derived.

Square numbers

A square number is the result of multiplying an integer by itself. For instance, taking an integer \(n\) and squaring it results in \(n^2\). This simple multiplication gives rise to interesting patterns and properties in mathematics, especially concerning odd and even numbers.

In the exercise, squaring the odd integer \((2n+1)\) produced the expression \(4n^2 + 4n + 1\), demonstrating that the square of any odd number will always follow a particular format, \(8m+1\).

Important points about square numbers include:

They can be even or odd, depending on the original integer.
Odd integers squared result in odd square numbers.
Square numbers grow rapidly compared to linear numbers, making them significant in algebra and geometry.

Understanding how odd squares fit into specific patterns helps in deeper studies of number theory and mathematical proofs.

91影视

Show that the square of every odd integer is of the form \(8 m+1\).

Short Answer

Step by step solution

Write the odd integer in the form of \(2n+1\)

Calculate the square of the odd integer

Simplify the expression

Rewrite the expression in the form \(8m+1\)

Conclude the proof

Key Concepts

Odd integers

Mathematical proof

Square numbers

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