Problem 40 - Precalculus with Limits

Question

In Exercises 25 - 30, prove the inequality for the indicated integer values of \( n \). A factor of \( \left(2^{2n + 1} + 1\right) \) is \( 3 \).

Step-by-Step Solution

Answer

Yes, 3 is a factor of \(2^{2n + 1} + 1\).

Step 1: Understand modular arithmetic

In modular arithmetic, we care about the remainder that's left when we divide by a certain number (in this case 3). So, our task is to confirm that \(2^{2n + 1} + 1\) leaves a remainder of 0 when divided by 3.

Step 2: Find pattern of powers of 2 modulo 3

First, find the remainder of powers of 2 divided by 3. They form a cycle: \(2^1 \equiv 2 (mod 3)\), \(2^2 \equiv 1 (mod 3)\), \(2^3 \equiv 2 (mod 3)\), \(2^4 \equiv 1 (mod 3)\), and so on. Notice that for even powers of 2, the remainder is always 1 when divided by 3.

Step 3: Apply the pattern to the given equation

Since the pattern repeats every 2 powers of 2, substitute \(2n+1\) with an odd number to find the mod of the expression \(2^{2n + 1} + 1\). This gives: \(2^{2n + 1} + 1 \equiv 2 + 1 \equiv 0 (mod 3)\)

Key Concepts

Powers of Integers

Understanding the powers of integers is crucial to grasping many mathematical concepts, including modular arithmetic. When we talk about powers of integers, we are referring to an integer raised to a certain exponent. For example, in the expression \(2^n\), 2 is the base, and \(n\) is the exponent.
This operation multiplies the base number by itself as many times as specified by the exponent. It's like repeated multiplication. Each increment of the exponent multiplies the base number one more time. For instance, \(2^3 = 2 \times 2 \times 2 = 8\).

• Exponent increases: As the exponent value rises, so does the overall value of the expression (for bases greater than one).
• Zero exponent: Any non-zero integer to the power of zero results in one, e.g., \(2^0 = 1\).
• Negative exponent: This represents a reciprocal, such as \(2^{-1} = \frac{1}{2}\).

Grasping the cyclical nature of powers is essential, especially in modular arithmetic, where we observe the repeated patterns over different modular bases.

Modulo Operation

The modulo operation is a concept from number theory, often symbolized as 'mod'. It refers to the remainder after division of one number by another. In simple terms, for any two positive integers \(a\) and \(b\), \(a \mod b\) is the remainder when \(a\) is divided by \(b\).
For example, if we divide 17 by 5, the division yields a quotient of 3 and a remainder of 2, hence \(17 \mod 5 = 2\). A common use of this in proofs is to confirm properties of numbers under different bases.

• Cyclical patterns: Numbers exhibit specific cycles when put in modular conditions, often simplifying complex series into predictable patterns.
• Applications: It's widely used in cryptography, computer science, and solving real-world problems.

In this case, understanding the mod operation helps in distinguishing how powers of numbers behave under division by specified integers, such as 3, as shown in the exercise to determine if \(2^{2n+1} + 1\) is divisible by 3.

Mathematical Proofs

Mathematical proofs are structured arguments that rigorously demonstrate a statement's validity. A proof generally involves logical reasoning and established mathematical principles to show that a statement must be true in all cases.
There are various methods of proving, such as direct proof, proof by induction, proof by contradiction, etc. Each method has its steps and needs careful application of logic.

• Direct proof: This method involves straightforward reasoning from known facts to arrive at the statement to be proven.
• Induction: This is used particularly to prove statements about integers, where the truth for \(n\) implies the truth for \(n+1\).

The task in the original exercise was to prove that 3 is a factor of \(2^{2n+1} + 1\). This involves the cyclical nature of powers of 2 modulo 3 (as presented in step two), where you see that odd powers of 2 give a remainder of 2, leading ultimately to the expression equating to zero in this modular system, confirming the division by 3. Using such logical sequences and structures fortifies mathematical truths beyond subjective intuition.