Problem 15 - Precalculus: Mathematical for Calculus

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 \( n \) because it simplifies to \( 1 \mod 2 \).

Step 1: Understand the Expression

The expression we want to analyze is \( n^2 - n + 41 \). We are tasked to show that this expression is odd for all natural numbers \( n \). An odd number can be expressed in the form of \( 2k + 1 \) where \( k \) is an integer.

Step 2: Analyze the Expression in Modulo Arithmetic

To determine if \( n^2 - n + 41 \) is odd, we check its parity using modulo 2 arithmetic. If the expression modulo 2 results in 1 for all \( n \), it is always odd.

Step 3: Simplify Using Modulo 2

Consider \( n^2 - n + 41 \mod 2 \):- \( n^2 \mod 2 = n \mod 2 \) since if \( n \equiv 0 \pmod{2} \), \( n^2 \equiv 0 \pmod{2} \), and if \( n \equiv 1 \pmod{2} \), \( n^2 \equiv 1 \pmod{2} \).- Therefore, \( n^2 - n \equiv n - n \equiv 0 \pmod{2} \).- So, \( n^2 - n + 41 \equiv 0 + 41 \equiv 41 \equiv 1 \pmod{2} \).Thus, \( n^2 - n + 41 \) is odd for all \( n \).

Key Concepts

Understanding Natural Numbers

Natural numbers are the set of positive integers starting from 1, continuing to infinity: 1, 2, 3, and so on. These are the numbers we most commonly use in our day-to-day counting. They include no fractions, decimals, or negatives. When working with mathematical expressions or operations involving natural numbers, you are assured that all results pertain to whole, non-negative integers.

Understanding the nature of natural numbers is crucial when faced with a mathematical problem asking for a property that holds for 'all natural numbers.' It means the expression or property must work starting from 1 upwards without skipping any number in this infinite set.

• Natural numbers are always positive and increment one at a time.
• They are the backbone of foundational mathematical operations.
• No negative or fractional numbers are included in natural numbers.

For the problem at hand, it means testing the expression across these numbers to prove its stated property.

The Role of Modulo Arithmetic

Modulo arithmetic, often referred as mod, is a way of finding the remainder when one number is divided by another. In the context of determining the parity (odd or even nature) of numbers, we commonly use mod 2 (or modulo 2). If a number is even, it leaves a remainder of 0 when divided by 2; if it's odd, the remainder is 1.

In this exercise, checking whether the expression \( n^2 - n + 41 \) is odd involves evaluating it under modulo 2 arithmetic. By simplifying expressions using mod 2:

• When \( n \) is even (\( n \equiv 0 \pmod{2} \)), both \( n^2 \equiv 0 \) and \( -n \equiv 0 \); thus \( n^2 - n \equiv 0 \).
• When \( n \) is odd (\( n \equiv 1 \pmod{2} \)), both \( n^2 \equiv 1 \) and \( -n \equiv 1 \); thus \( n^2 - n \equiv 0 \).
• The term 41 simplifies to \( 1 \pmod{2} \) because the same rule applies.
  The entire expression \( n^2 - n + 41 \equiv 1 \), showing that it is indeed odd.

Modulo arithmetic offers a powerful tool in simplifying and proving properties like parity efficiently, ensuring complex numerical patterns are easily identified.

Understanding Odd Numbers

An odd number is any integer of the form \( 2k + 1 \), where \( k \) is an integer. This form represents any number having an odd parity, different from even numbers, which can be expressed as \( 2k \).

Odd numbers are peculiar in their sequence as they always maintain an interval difference of two: 1, 3, 5, 7, and so forth. These properties arise naturally from their definition:

• Each odd number when divided by 2 leaves a remainder of 1.
• They persist throughout the entirety of the integer set.
• In mathematical expressions, their presence and manipulation can affect outcomes significantly.

In the exercise, showing that \( n^2 - n + 41 \) is always odd for natural numbers involves understanding how each part of the expression configures to result as distinctively odd. This highlights how algebraic expressions can consistently produce certain parities, using basic number properties.