Question

Use the method of direct proof to prove the following statements. If \(a\) is an odd integer, then \(a^{2}+3 a+5\) is odd.

Short answer

By defining an odd number as \(2n + 1\), substituting it in the given expression, and simplifying, it is established that \(a^{2}+3a+5\) is an odd number.

Step-by-step solution

Define Odd Number

An odd number can be expressed as \(2n + 1\), where \(n\) is an integer. If \(a\) is an odd number, then \(a = 2n + 1\). Substitute this into the original expression.

Inductive Substitution

Substitute \(a = 2n + 1\) into the expression \(a^{2}+3a+5\) to obtain \((2n+1)^{2}+3(2n+1)+5\). This simplifies to \(4n^{2} + 2n + 1 + 6n + 3 + 5\), which further simplifies to \(4n^{2} + 8n + 9\).

Demonstrating Oddness

The expression \(4n^{2} + 8n + 9\) can be written as \(2(2n^{2} + 4n + 4) + 1\), where \(2n^{2} + 4n + 4\) is an integer. Hence the expression is of the form \(2m+1\), making it an odd number.

Key concepts

Understanding Odd Integers

An odd integer is a number that is not divisible by 2, and when divided by 2, it leaves a remainder of 1. In formal mathematical terms, an odd number can be written as \(2n + 1\), where \(n\) is an integer (a whole number, not a fraction). This representation is incredibly useful when performing algebraic manipulations, as it encapsulates the essence of what makes a number odd within a variable framework.

For instance, if we consider \(a\) to be an odd integer, we can say \(a = 2n + 1\) where \(n\) stands for any integer value. This formula becomes a starting point to explore how odd numbers behave when used in different mathematical operations such as squaring, adding or multiplying—a common inquiry in various proofs and exercises. Understanding this is crucial, as it is a stepping stone in grasping the nature of many mathematical theorems and problems dealing with integers.

Inductive Substitution in Action

Inductive substitution is a strategy in algebra where we replace a variable with an expression that represents it. This is often used to simplify complex problems or to prove certain properties about the expressions involved. In the case of our exercise, inductive substitution involves replacing \(a\) with \(2n + 1\) in the expression \(a^{2} + 3a + 5\).

The beauty of inductive substitution lies in its ability to make patterns evident. Once we perform the substitution, we obtain a new expression \(4n^{2} + 8n + 9\), which on the surface, may not immediately reveal its odd or even nature. However, by restructuring it as \(2(2n^{2} + 4n + 4) + 1\), it becomes clear that the result is an odd number since it's in the form \(2m + 1\) where \(m\) is another integer. This method showcases the transformative power of substituting expressions to unveil underlying mathematical truths. It's a prime example of using a systematic approach to simplify and resolve algebraic problems, making them more digestible.

The Essence of Mathematical Proof

Mathematical proof is the cornerstone of establishing truth within mathematics. It is a logical argument that confirms the validity of a mathematical statement beyond any doubt. In the language of mathematics, a proof must show that a statement is always true, under the given assumptions and axioms of the system. Direct proof, which is used in our exercise, is a straightforward method where we start with known truths and use logical steps to show that another statement must also be true.

For our problem, we prove that if \(a\) is an odd integer, then the expression \(a^{2}+3a+5\) is also odd. The direct proof begins with the assumption that \(a\) is odd, expressed as \(2n + 1\), and through a series of unquestionable algebraic steps, we show the final expression also takes the form of an odd integer, \(2m + 1\), hence proving the original claim. Proofs like this are central to the discipline of mathematics, as they assure us of the reliability and consistency of mathematical principles and allow us to build upon known foundations with confidence.