Problem 17 - Precalculus: Mathematics for Calculus

Question

\(n^{2}-n+41\) 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 Problem

We need to prove that the expression \(n^2 - n + 41\) is odd for all natural numbers \(n\). This means regardless of the value of \(n\), the expression should result in an odd number.

Step 2: Check Odd and Even Cases

A number is odd if its remainder is 1 when divided by 2. Examine the expression for both odd and even values of \(n\) to see if \(n^2 - n + 41\) is always odd.

Step 3: Consider Even Values of n

Let \(n = 2k\), where \(k\) is a natural number. Then: \[n^2 = (2k)^2 = 4k^2\]Subtract \(n = 2k\) to get: \[n^2 - n = 4k^2 - 2k\]Add 41: \[n^2 - n + 41 = 4k^2 - 2k + 41\]Notice that \(4k^2 - 2k\) is even; thus, when added to 41, which is odd, the result is odd (even + odd = odd).

Step 4: Consider Odd Values of n

Let \(n = 2k + 1\), where \(k\) is a natural number. Then: \[n^2 = (2k + 1)^2 = 4k^2 + 4k + 1\]Subtract \(n = 2k + 1\) to get: \[n^2 - n = 4k^2 + 4k + 1 - (2k + 1) = 4k^2 + 2k\]Add 41: \[n^2 - n + 41 = 4k^2 + 2k + 41\]Here, \(4k^2 + 2k\) is even; thus when added to 41, the result remains odd (even + odd = odd).

Step 5: Conclusion

For any natural number \(n\), whether \(n\) is odd or even, \(n^2 - n + 41\) results in an odd number. Thus, the expression is guaranteed to be odd for all natural numbers \(n\).

Key Concepts

Odd and Even Numbers

Odd and even numbers are fundamental concepts in arithmetic and number theory. They classify integers based on their divisibility by 2. An even number is any integer that can be divided by 2 without a remainder, like 2, 4, 6, 8, and so on. This can be expressed mathematically as any number of the form \( n = 2k \), where \( k \) is an integer.

Odd numbers, on the other hand, leave a remainder of 1 when divided by 2. Some common examples include 1, 3, 5, 7, and so on. Mathematically, any number of the form \( n = 2k + 1 \) is considered odd. One interesting property to remember is that when you add an even and an odd number together, the result is always odd. The exercise determined that the polynomial expression we evaluated maintained this property.

• Even numbers: \( n = 2k \)
• Odd numbers: \( n = 2k + 1 \)
• Even + Odd = Odd

Understanding this helps in recognizing why certain operations in our specific polynomial will yield either odd or even results, which is crucial to solving such problems.

Natural Numbers

Natural numbers are the set of positive integers beginning from 1, and they continue eternally upwards (1, 2, 3, 4, ...). They are the numbers we naturally use when counting objects or entities. Natural numbers do not include zero in the traditional sense, and negative numbers are explicitly absent.

In algebra and number theory, these numbers often serve as inputs or variables in equations and expressions. When we say we are considering all natural numbers \( n \) in an equation, we mean that \( n \) can take any value from 1 to infinity. This concept is essential in understanding why, in the problem, all natural numbers are evaluated to see the effect on the expression \( n^2 - n + 41 \).

• Starts from 1
• Extends infinitely
• No negative numbers

Knowing this is crucial because we need to ensure the expression always gives a certain type of number (odd, in this case) across the entire domain of natural numbers.

Polynomial Expressions

Polynomial expressions are made up of variables raised to whole number powers and multiplied by coefficients. The expression given in the exercise, \( n^2 - n + 41 \), is a simple polynomial of degree 2. This expression includes several basic elements of polynomials:

• Terms: Individual parts of the expression. Here, \( n^2 \), \( -n \), and 41.
• Coefficients: Numbers that multiply the variable terms (implicitly 1 for \( n^2 \) and -1 for \( -n \)).
• Constant: A standalone number without a variable, in this context 41.
• Degree: The largest exponent in the polynomial, which is 2 for this expression (due to \( n^2 \)).

These particular elements of the polynomial affect how it behaves as \( n \) changes and why it consistently becomes odd. By breaking it into parts and examining even and odd entries separately, we confirm its behavior in relation to odd and even number rules.