Problem 37 - Precalculus Mathematics for Calculus

Question

True or False? Determine whether each statement is true or false. If you think the statement is true, prove it. If you think it is false, give an example in which it fails. (a) \(p(n)=n^{2}-n+11\) is prime for all \(n\) (b) \(n^{2}>n\) for all \(n \geq 2\) (c) \(2^{2 n+1}+1\) is divisible by 3 for all \(n \geq 1\) (d) \(n^{3} \geq(n+1)^{2}\) for all \(n \geq 2\) (e) \(n^{3}-n\) is divisible by 3 for all \(n \geq 2\) (f) \(n^{3}-6 n^{2}+11 n\) is divisible by 6 for all \(n \geq 1\)

Step-by-Step Solution

Answer

(a) False, (b) True, (c) True, (d) False, (e) True, (f) True.

Step 1: Analyze Statement (a)

Consider the expression \( p(n) = n^2 - n + 11 \). We need to determine if it is prime for all \( n \). Substitute \( n = 5 \): \[p(5) = 5^2 - 5 + 11 = 25 - 5 + 11 = 31\]This is prime. Try another value, like \( n = 11 \):\[p(11) = 11^2 - 11 + 11 = 121 - 11 + 11 = 121\]121 is not a prime number (since \( 121 = 11 \times 11 \)). Therefore, the statement is false.

Step 2: Analyze Statement (b)

Consider if \( n^2 > n \) for all \( n \geq 2 \). Try \( n = 2 \):\[2^2 = 4 > 2\]This holds true. For any larger \( n \), \( n^2 \) grows more due to the squaring, hence \( n^2 > n \) for \( n \geq 2 \). So, the statement is true.

Step 3: Analyze Statement (c)

Consider \( 2^{2n+1} + 1 \) and its divisibility by 3 for all \( n \geq 1 \). Start with \( n = 1 \):\[2^3 + 1 = 8 + 1 = 9\]9 is divisible by 3. Now try \( n = 2 \):\[2^5 + 1 = 32 + 1 = 33\]33 is also divisible by 3. This pattern holds due to properties of powers of 2. Thus, the statement is true.

Step 4: Analyze Statement (d)

We need to check if \( n^3 \geq (n+1)^2 \) for \( n \geq 2 \). Start with \( n = 2 \):\[2^3 = 8 \]\[(2+1)^2 = 3^2 = 9\]8 is not greater than or equal to 9; thus it fails at \( n = 2 \). Therefore, the statement is false.

Step 5: Analyze Statement (e)

Check if \( n^3 - n \) is divisible by 3 for \( n \geq 2 \). Consider:For \( n = 2 \):\[2^3 - 2 = 8 - 2 = 6\]6 is divisible by 3.For \( n = 3 \):\[3^3 - 3 = 27 - 3 = 24\]24 is divisible by 3.This follows due to the structure of \( n(n^2 - 1) \), where \( n^2 - 1 \) always produces a multiple of 3 for consecutive integers, making \( n^3 - n \) divisible by 3. Therefore, this statement is true.

Step 6: Analyze Statement (f)

Check if \( n^3 - 6n^2 + 11n \) is divisible by 6 for \( n \geq 1 \). Simplify this by checking modular arithmetic:**Modulo 2:** All terms are even, hence divisible by 2.**Modulo 3:**Let's test it directly. Start with \( n = 1 \):\[1^3 - 6 \times 1^2 + 11 \times 1 = 1 - 6 + 11 = 6\]6 is divisible by 6. Try \( n = 2 \):\[2^3 - 6 \times 2^2 + 11 \times 2 = 8 - 24 + 22 = 6\] 6 is divisible by 6. The polynomial produces zero or divisible outcomes by 6 in its similar progressive structure values for all integers, confirming this statement is true.

Key Concepts

Prime Numbers

Prime numbers are special numbers greater than 1 that have no divisors other than 1 and themselves. They are important because they cannot be broken down into other numbers through multiplication, acting as the building blocks for other numbers. For example, the number 5 is prime because the only divisibility factors for 5 are 1 and 5 itself.

• Prime number examples: 2, 3, 5, 7, 11, 13, etc.
• All even numbers greater than 2 are not prime, because they have at least three divisors: 1, 2, and themselves.
• To check if a number is prime, ensure it is not divisible by any integer other than 1 and itself.

Polynomial expressions like \(p(n)=n^2-n+11\) aim to present prime numbers only, but as shown in the step-by-step solution, substituting certain values of \(n\) like 11 results in non-prime numbers. It proves that this specific polynomial does not produce a prime number for all \(n\). Understanding why certain values produce non-prime outcomes helps clarify the pattern and nature of polynomials in relation to primes.

Divisibility

Divisibility describes if one number can be divided by another without leaving a remainder. For instance, 10 is divisible by 5 because 10 divided by 5 equals 2 with no remainder. In mathematics, proving divisibility allows us to predict and manipulate situations easily.

• Number 6 is divisible by both 3 and 2.
• An integer \(a\) is divisible by another integer \(b\) if there exists an integer \(c\) such that \(a = bc\).
• Divisibility can be tested using examples or through modular arithmetic.

In the exercise provided, the polynomial expressions like \(2^{2n+1}+1\) consistently demonstrate divisibility by 3 for certain values. This can be verified using prime factor properties and testing several basic values of \(n\). Realizing why a polynomial function behaves in this way underlies the importance of recognizing patterns in number divisibility.

Inequalities

Inequalities show relationships between values or expressions that are not equivalent. Notation such as \(<, >, \leq, \geq\) is used to express inequalities, signifying less than, greater than, less than or equal to, and greater than or equal to respectively. Understanding inequalities is key to solving many algebraic problems.

• \(n^2 > n\) for all \(n \geq 2\) illustrates an inequality showing that a squared number grows faster than the number itself.
• Solving inequalities can involve adding, subtracting, multiplying or dividing both sides of the inequality by the same number, similar to equations.
• While analyzing polynomial expressions such as \(n^3 \geq (n+1)^2\), inequalities help determine value ranges or validate conditions.

Analyzing inequalities help in understanding limits and relationships of functions and sequences. They provide reasoning behind the step exercise results, testing constraints like \(n^3 \geq (n+1)^2\) that influence polynomial behaviors.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, involving terms like constants, powers, and sums. They capture numerous mathematical phenomena and lead to understanding complex algebraic relationships.

• Forms can vary from \(a_nx^n+...+a_1x+a_0\), meaning terms like \(x^2, x^3\) can appear with associated degrees.
• The degree of a polynomial is the highest power of the variable, such as \(x^3\) in \(n^3-6n^2+11n\).
• Determining factors for divisibility or roots can be done through techniques like factoring or evaluating specific points.

In expression \(n^3-6n^2+11n\), polynomials reveal conditions for divisibility by 6. Recognizing patterns like the constant occurrence of zero helps verify divisibility criteria when tested through several values of \(n\). Polynomial functions highlight intricate relations between algebraic structures.