/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 The difference of the squares of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 54

The difference of the squares of any two consecutive integers is odd.

Short Answer

Expert verified
The difference of the squares of any two consecutive integers is odd. We denoted the first integer as 'n' and the consecutive integer as '(n+1)'. We computed the squares of both integers and calculated the difference as \((n+1)^2 - n^2\). After simplifying the expression, we got '2n + 1', which is an odd integer. Hence, the statement is proven to be true.

Step by step solution

01

Define the consecutive integers.

Let's denote the first integer as 'n', then the consecutive integer will be '(n+1)'. These are two consecutive integers, where 'n' is an arbitrary integer.
02

Compute the squares of the integers.

We need to compute the squares of both integers. The square of the first integer 'n' will be \(n^2\), and the square of the consecutive integer '(n+1)' will be \((n+1)^2\).
03

Compute the difference of the squares.

Now, we need to calculate the difference between the squares of these consecutive integers. The difference will be \((n+1)^2 - n^2\).
04

Simplify the expression.

Let's simplify the expression we got in step 3. We have: \[ \begin{aligned} (n+1)^2 - n^2 &= (n^2+2n+1)-n^2 \\ &= (n^2 - n^2) + 2n + 1 \\ &= 2n + 1 \end{aligned} \]
05

Analyze the result.

Observe that the result, '2n+1', is the product of an even integer '2n' and an odd integer '1', which results in an even integer. By adding 1 to an even integer, we always obtain an odd integer. Therefore, the difference of the squares of any two consecutive integers is odd. So, we have successfully proved that the difference of the squares of any two consecutive integers is odd.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Consecutive Integers
In mathematics, consecutive integers are numbers that follow each other in order, without any gaps. Simply put, they are numbers that come one after another. When we define consecutive integers, we often start with a variable, say, "n".
For example, if "n" represents your first integer, then "n+1" would naturally follow as the next consecutive integer.
  • If "n" is 3, then the consecutive integer is 4.
  • If "n" is 10, then the consecutive integer is 11.
Consecutive integers are useful in various mathematical concepts because of their straightforward and predictable nature. This makes them particularly handy when proving theorems or exploring patterns, such as calculating the difference of their squares.
Difference of Squares
The difference of squares is a significant concept in mathematics. It describes the result of subtracting the square of one number from the square of another number. A specific interesting case is the difference of squares of consecutive integers.
When dealing with two consecutive integers, "n" and "n+1", the formula for their difference of squares becomes:
\[(n+1)^2 - n^2\]
Upon expanding this, it simplifies to\[2n + 1\]
This shows a neat property: the outcome changes at every integer step, which is crucial in understanding patterns generated by numbers.
  • The term "2n" always produces an even result, as multiplying any integer by 2 yields an even number.
  • Adding 1 makes the entire expression odd, ensuring that every difference of squares of consecutive integers is odd.
Mathematical Proof
A mathematical proof is a logical process used to demonstrate the truth of a statement beyond any doubt. When proving the property about the difference of squares of consecutive integers, we use algebraic manipulation to reach our conclusion.
The goal here is to show the expression \[(n+1)^2 - n^2\]
always results in an odd number, no matter what integer "n" is chosen.
Here's the proof in brief steps:
  • Define your integers, "n" and "n+1".
  • Calculate their squares: "n^2" and "(n+1)^2."
  • Subtract the first square from the second, resulting in:\[(n^2+2n+1) - n^2 = 2n + 1\]
  • The term "2n" being even added to 1 is always odd; hence, the statement is true.
Such proof is essential because it stands as a universal statement across all integer values. This approach is a cornerstone of discrete mathematics, allowing us to confirm patterns and truths in a systematic way.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

How many zeros are at the end of \(45^{8} \cdot 88^{5}\) ? Explain how you can answer this question without actually computing the number. (Hint: \(10=2 \cdot 5 .\) )

Prove that for all integers \(n, n^{2}-n+3\) is odd.

There are distinct integers \(m\) and \(n\) such that \(\frac{1}{m}+\frac{1}{n}\) is an integer.

Each of the statements in \(20-23\) is true. For each, (a) rewrite the statement using a variable or variables and the form \(V\) if ___ then ___ and (b) write the first sentence of a proof (the "starting point") and the last sentence of a proof (the "conclusion to be shown"). Note that you do not need to understand the statements in order to be able to do these exercises. 20\. For all integers \(m\), if \(m>1\) then \(0<\frac{1}{m}<1\).

Consider the statement "For all integers \(n\), if \(n^{2}\) is odd then \(n\) is odd." a. Write what you would suppose and what you would need to show to prove this statement by contradiction. b. Write what you would suppose and what you would nced to show to prove this statement by contraposition.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.