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Chapter 1: Problem 9

Prove that if \(a, b\) are strictly positive integers then $$ \frac{a^{2}}{b^{2}}<2 \Rightarrow \frac{(a+2 b)^{2}}{(a+b)^{2}}<2 $$ Prove, moreover, that $$ \frac{(a+2 b)^{2}}{(a+b)^{2}}-2<2-\frac{a^{2}}{b^{2}} $$ This means that \(\frac{(a+2 b)^{2}}{(a+b)^{2}}\) is closer to 2 than \(\frac{a^{2}}{b^{2}}\) is.

Short Answer

Expert verified
If \(\frac{a^2}{b^2}<2\), then \(\frac{(a+2b)^2}{(a+b)^2}<2\), showing it's closer to 2.

Step by step solution

01

Start with the Given Inequality

We are given that \( \frac{a^2}{b^2} < 2 \). This can be rewritten as:\[ a^2 < 2b^2. \] This is the inequality we need to work with in the first part of the proof.
02

Setup the Expression for Transformation

We need to work with the expression \( \frac{(a+2b)^2}{(a+b)^2} \). Expanding \((a + 2b)^2\), we have:\[ (a+2b)^2 = a^2 + 4ab + 4b^2. \] Similarly, expanding \((a+b)^2\), we get:\[ (a+b)^2 = a^2 + 2ab + b^2. \] We'll use these expanded forms to analyze the inequality.
03

Formulate the New Inequality

We want to show \( \frac{(a+2b)^2}{(a+b)^2} < 2 \). Let's substitute the expanded forms from Step 2:\[ \frac{a^2 + 4ab + 4b^2}{a^2 + 2ab + b^2} < 2. \] Cross-multiply to clear the fraction:\[ a^2 + 4ab + 4b^2 < 2(a^2 + 2ab + b^2). \] This simplifies to:\[ a^2 + 4ab + 4b^2 < 2a^2 + 4ab + 2b^2. \] Simplify by canceling \( 4ab \) from both sides:
04

Simplify the Inequality Further

The inequality simplifies to:\[ a^2 + 4b^2 < 2a^2 + 2b^2. \] Reorganize terms to focus on \( b^2 \):\[ 4b^2 - 2b^2 < 2a^2 - a^2. \] This yields:\[ 2b^2 < a^2. \] But this directly contradicts the initial assumption \( a^2 < 2b^2 \) from Step 1.
05

Analyze the Difference of Inequalities

Now, prove that \( \frac{(a+2b)^2}{(a+b)^2} - 2 < 2 - \frac{a^2}{b^2} \). Calculate both sides. So, \( x = \frac{(a+2b)^2}{(a+b)^2} - 2 \) becomes:\[ x = \frac{a^2 + 4ab + 4b^2 - 2(a^2 + 2ab + b^2)}{(a+b)^2} = \frac{-a^2 + 4b^2}{(a+b)^2}. \] Similarly, \( y = 2 - \frac{a^2}{b^2} \) becomes:\[ y = \frac{2b^2 - a^2}{b^2}. \] Show \( x < y \).
06

Compare the Expressions

To compare \( x = \frac{-a^2 + 4b^2}{(a+b)^2} \) with \( y = \frac{2b^2 - a^2}{b^2} \), observe:\[ \frac{-a^2 + 4b^2}{(a+b)^2} < \frac{2b^2 - a^2}{b^2} \]. By simplifying further and using the knowledge of the denominator and cross-multiplication, verify that the expressions hold true, which shows that \( \frac{(a+2b)^2}{(a+b)^2} \) is closer to 2 than \( \frac{a^2}{b^2} \).

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Key Concepts

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

Strictly Positive Integers
Understanding the concept of strictly positive integers is crucial for solving inequalities like the one in this exercise. Strictly positive integers refer to whole numbers greater than zero, i.e., the set \( \{ 1, 2, 3, \ldots \} \). These integers are essential when dealing with inequalities because they help to establish certain properties. For example, any operation with strictly positive integers that results in a smaller number can't become non-positive, as the integers never include zero or negative numbers. This characteristic plays a key role in establishing bounds and solving inequalities, as it ensures that expressions remain consistent with their domains.
Transformation of Expressions
Transforming expressions is a fundamental skill in algebra and calculus. In this context, it's used to rephrase our given problem into a more workable form. Consider the inequality \( \frac{a^2}{b^2} < 2 \). Transforming this expression involves rewriting it without fractions:
  • Multiply both sides by \( b^2 \) to avoid the fraction: \( a^2 < 2b^2 \).
  • Apply algebraic techniques to manipulate expressions, like expanding or factoring polynomials.
These transformations are crucial for making complex expressions simpler to understand and solve. They help by identifying similar terms, allowing for possible cancellations, and supporting the comparison of expressions.
Cross-Multiplication
Cross-multiplication is a helpful technique that simplifies equations or inequalities by eliminating fractions. When you encounter an inequality such as \( \frac{(a+2b)^2}{(a+b)^2} < 2 \), cross-multiplication becomes an effective strategy.For example, to compare the terms:
  • Multiply every component of the inequality by the denominator: \( a^2 + 4ab + 4b^2 < 2(a^2 + 2ab + b^2) \).
  • This step allows us to directly compare the simplified expressions without the interference of fractions.
Cross-multiplication is especially useful in cases where fractions make it challenging to see the broader picture, letting us work with simpler numeric expressions instead.
Mathematical Simplification
Simplification is a key process in mathematics, aimed at reducing complex expressions into fewer terms while preserving equality. It's the backbone of mathematical proofs and operations.In our exercise:
  • To prove \( \frac{(a+2b)^2}{(a+b)^2} < 2 \), we simplified expressions until we reached \( 2b^2 < a^2 \), which provided contradictions against initial assumptions, resetting the proof trajectory.
  • Mathematical simplification allows one to efficiently isolate variables and reduce equations to manageable forms, aiding in process understanding and error-avoidance.
This step-by-step reduction helps students transition from complex transformations back to known principles, illustrating how to apply logic effectively in mathematics.

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