Problem 17 - Precalculus Mathematics 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 remains odd whether \(n\) is even or odd.

Step 1: Understanding the Expression

We need to show that the expression \(n^2 - n + 41\) is odd for all natural numbers \(n\). An odd number can be expressed as \(2k + 1\) where \(k\) is an integer.

Step 2: Analyzing Parity of \(n\)

Natural numbers \(n\) can either be odd or even. We'll consider both cases separately to check if \(n^2 - n + 41\) remains odd.

Step 3: Case 1: \(n\) is Even

Let's assume \(n = 2k\) for some integer \(k\). Then, \(n^2 = (2k)^2 = 4k^2\) and \(n = 2k\). Substituting these into the expression gives us \(n^2 - n + 41 = 4k^2 - 2k + 41\). Each of \(4k^2\) and \(-2k\) is even, therefore their sum \(4k^2 - 2k\) is even, which means \(4k^2 - 2k + 41\) is odd because 41 is odd.

Step 4: Case 2: \(n\) is Odd

Now assume \(n = 2k + 1\) for some integer \(k\). Then, \(n^2 = (2k + 1)^2 = 4k^2 + 4k + 1\) and \(n = 2k + 1\). Substituting into the expression gives \(n^2 - n + 41 = (4k^2 + 4k + 1) - (2k + 1) + 41\). Simplifying, we have \(4k^2 + 2k + 41 \). Here, \(4k^2 + 2k\) is even, making the entire expression odd because 41 is odd.

Step 5: Conclusion

In either case, whether \(n\) is even or odd, \(n^2 - n + 41\) remains an odd number. Therefore, the expression is odd for all natural numbers \(n\).

Key Concepts

Parity Analysis

Parity analysis involves examining whether a number is odd or even. When analyzing an expression such as \(n^2 - n + 41\), we explore how the expression behaves based on whether \(n\) is odd or even.

• An expression's parity (oddness or evenness) depends on its individual terms and their arithmetic operations.
• Addition or subtraction of two even numbers results in an even number.
• Similarly, adding or subtracting an odd number with an even number give us an odd result.

To determine the parity of the complex expression \(n^2 - n + 41\), we delve into specifics about each term’s parity in different contexts. This helps identify consistent patterns, ensuring the outcome of the expression remains odd for any natural number input.

Odd and Even Numbers

Odd and even numbers are fundamental concepts in mathematics. An even number can be divided by 2 without leaving a remainder, such as 2, 4, 6, etc. Odd numbers, like 1, 3, 5, leave a remainder of 1 when divided by 2.

• An even number can be represented in the form \(2k\), where \(k\) is an integer.
• An odd number is represented as \(2k + 1\).

These simple definitions help us assess the nature of more complex expressions. When determining the parity of \(n^2 - n + 41\), identifying whether \(n\) itself is an odd or even number dictates the nature of each term. For example, if \(n = 2k\) (even), then \(n^2\) and \(n\) are both even, simplifying analysis. If \(n = 2k+1\) (odd), these same terms transform, but \(41\), being odd, largely influences the final outcome.

Natural Numbers

Natural numbers are the set of positive integers starting from 1 (i.e., 1, 2, 3, ...). This set does not include zero or any negative numbers. In mathematics and real-world applications, natural numbers are most commonly used for counting and ordering.When analyzing expressions like \(n^2 - n + 41\), the base assumption is that \(n\) represents a natural number. This allows us to limit our consideration of values to only positive whole numbers.

• The property of being a natural number ensures that \(n\) is consistently either odd or even.
• This consistent nature simplifies predictions on the parity of expressions like \(n^2 - n + 41\).

Since natural numbers inherently encompass both odd and even numbers, their application ensures comprehensive analysis of any expression reliant on the parity principles we've discussed. The focus on natural numbers helps to affirm the oddness in this particular expression across all possible instances of \(n\).