Problem 17 - Algebra and Trigonometry

Question

Show that \(n^{2}-n+41\) is odd for all natural numbers \(n\)

Step-by-Step Solution

Answer

The expression \(n^2 - n + 41\) is odd for all natural numbers \(n\).

Step 1: Understand the Expression

The expression in question is \(n^2 - n + 41\). We need to determine whether this expression is always odd for any natural number \(n\).

Step 2: Analyze Parity of Natural Numbers

Natural numbers can either be even or odd. We will examine both cases to determine the parity of \(n^2 - n + 41\).

Step 3: Examine the Case when n is Even

If \(n\) is even, then \(n = 2k\) for some integer \(k\). Substituting this into the expression gives us \(n^2 - n + 41 = (2k)^2 - 2k + 41 = 4k^2 - 2k + 41\). This expression is odd because \(4k^2\) is even, \(-2k\) is even, and adding 41 (an odd number) results in an odd value.

Step 4: Examine the Case when n is Odd

If \(n\) is odd, then \(n = 2k + 1\) for some integer \(k\). Substituting this into the expression gives us \(n^2 - n + 41 = (2k + 1)^2 - (2k + 1) + 41 = 4k^2 + 4k + 1 - 2k - 1 + 41 = 4k^2 + 2k + 41\). Here, \(4k^2\) is even, \(2k\) is even, and adding 41 results in an odd value.

Step 5: Conclusion

For both even and odd natural numbers \(n\), the expression \(n^2 - n + 41\) always evaluates to an odd number because it combines even components with the odd number 41, which results in an odd number as the sum.

Key Concepts

Natural Numbers

Natural numbers are the set of numbers used for counting and ordering. They begin from 1 and go upwards to infinity, without any fractional or decimal parts. Here's why natural numbers are important:

• **Basic Operations:** With natural numbers, you can perform basic operations like addition, subtraction, multiplication, and sometimes division (when it results in another natural number).
• **Foundation in Mathematics:** They are foundational in many areas of mathematics. Natural numbers are often the first numbers students learn about.
• **Set Representation:** In set theory, natural numbers are often denoted by the letter **\(\mathbb{N}\)**.

When we consider expressions and mathematical equations, natural numbers help us understand values in the simplest, unbroken form possible. Remember, every natural number is a whole number, but not every whole number is a natural number (since whole numbers include zero).

Parity

Parity is a term used to describe whether an integer is even or odd. Understanding parity helps us analyze the behavior of mathematical expressions. Here's what you need to know:

• **Even Numbers:** These are numbers divisible by 2, like 2, 4, 6, etc. Algebraically, an even number can be expressed as \(2k\), where \(k\) is an integer.
• **Odd Numbers:** These are numbers not divisible by 2, like 1, 3, 5, etc. An odd number can be represented as \(2k+1\), where \(k\) is an integer.
• **Expression Analysis:** When working with expressions like \(n^2 - n + 41\), analyzing parity involves breaking down each term to see how they contribute to the overall parity of the expression.

Using parity, we can predict the result of expressions based on the values of \(n\). For example, adding an odd number \(41\) to any even result will yield an odd number, demonstrating the role of parity in algebraic expressions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operation symbols. Understanding these expressions allows us to solve equations and find unknown values.

• **Components:** An algebraic expression is composed of terms, coefficients, and variables. Terms are separated by plus or minus signs.
• **Example Expression:** In \(n^2 - n + 41\), there are three terms: \(n^2\), \(-n\), and \(+41\). Each term can influence the output differently.
• **Simplification and Evaluation:** By substituting values for \(n\), like in the case where \(n\) could be even or odd, one can evaluate the expression to understand its behavior. Simplification often involves combining like terms or factoring.

Algebraic expressions are vital for representing real-world situations in mathematical terms, allowing us to generalize problems and systematically find solutions.