Question

\(2^{n}-1\) gives the set of all odd natural numbers for all \(\mathrm{n} \in \mathrm{N}\). Comment on the given statement. (1) True for all values of \(\mathrm{n}\) (2) False (3) True for only odd values of \(\mathrm{n}\) (4) True for only prime values of \(\mathrm{n}\)

Short answer

Question: The expression \(2^n - 1\) gives all odd natural numbers. Comment on the given statement. Answer: (1) True for all values of \(n\) Explanation: Our analysis confirms that the expression \(2^n - 1\) indeed generates all odd natural numbers for every value of \(n\).

Step-by-step solution

Analyzing the expression for odd numbers

An odd number is any integer that can be represented by \(2k + 1\), where \(k\) is also an integer. Let's plug in some values of \(n\) in the expression \(2^n - 1\) and see whether we can generate all odd numbers. If we can generate all odd numbers, it will prove that the statement is true for all values of \(n\).

Test various values of \(n\)

Let's test various values of \(n\) to see what happens: \(n = 1\): \(2^1 - 1 = 1\) (odd) \(n = 2\): \(2^2 - 1 = 3\) (odd) \(n = 3\): \(2^3 - 1 = 7\) (odd) \(n = 4\): \(2^4 - 1 = 15\) (odd) No matter which natural number we substitute for \(n\), we get odd numbers. But we still haven't confirmed whether the expression generates every odd number.

Reversing the expression to prove all odd numbers can be formed

Now, we want to prove that every odd number can be formed by this expression by reversing the expression. We want \(2k + 1 = 2^n - 1\) for some integer \(k\). Solving for \(n\): \(2^n = 2k + 2\) \(n = \log_2{(2k + 2)}\) Now, we can see that for every odd number (\(2k + 1\)), there exists a value of \(n\) such that \(2n = 2k + 2\), which is a natural number.

Compare with the answer choices

Now, let's compare our findings with the answer choices: (1) True for all values of \(\mathrm{n}\): This is confirmed by our analysis, as we were able to generate all odd numbers using this expression. (2) False: This is not true, as we were able to confirm that the expression correctly generates all odd numbers. (3) True for only odd values of \(\mathrm{n}\): This is not true, as even when \(n\) is even, we still generated odd numbers. (4) True for only prime values of \(\mathrm{n}\): This is also not true, as we were able to generate odd numbers regardless of whether \(n\) is prime or not. As a result, the correct answer is (1) True for all values of \(n\).

Key concepts

Odd Numbers

Odd numbers are unique in their form as they differ from even numbers. An odd number can be expressed as \(2k + 1\), where \(k\) is an integer. This representation ensures that the number does not divide perfectly by 2, leaving a remainder of 1. Odd numbers form a limitless set of numbers: 1, 3, 5, 7, and so on. These numbers are essential in various branches of mathematics and often require special consideration in problems that involve divisibility, sequences, and series.

• Odd numbers always alternate with even numbers.
• When two odd numbers are added or subtracted, the result is even.
• Multiplying two odd numbers results in another odd number.

Recognizing and understanding odd numbers are vital in mathematical reasoning, helping solve complex problems involving sets and sequences.

Number Theory

Number theory is a branch of pure mathematics that deals primarily with integer numbers and their properties. It's often referred to as "higher arithmetic" and includes the study of prime numbers, divisors, and the solutions to equations in integers. One intriguing aspect of number theory is its focus on understanding the structures and behaviors of numbers under operations like addition, multiplication, and exponentiation.

• Prime numbers—the building blocks of integers—are a fundamental topic in number theory.
• Number theory is applied in cryptography, specifically for secure communications.
• It delves into questions about numbers that are simple to state but can be incredibly challenging to solve.

The statement from the exercise reflects number theory by exploring the nature of exponential expressions and their ability to generate numerical sequences, specifically odd numbers.

Logarithms

Logarithms are the inverses of exponents and provide an alternative way of expressing exponential relationships. When solving for an exponent in an equation, logarithms are used to "bring down" the exponent to make it more manageable. For example, given the expression \(2^n = 8\), the equivalent logarithmic form is \(n = \log_2{8}\).

• A logarithm answers the question: "To what exponent must the base be raised to obtain a certain number?"
• Common bases for logarithms are 10 (common logarithms) and \(e\) (natural logarithms).
• Logarithms have rules, such as the product rule, the quotient rule, and the power rule, which all simplify complex operations.

Logarithms help in understanding the generation of odd numbers in the original exercise, where solving for \(n\) using logarithms proves the expression \(2^n - 1\) covers all odd numbers.

Exponents

Exponents describe how many times a number, known as the base, is multiplied by itself. The notation \(b^n\) is read as "\(b\) raised to the power of \(n\)," indicating that \(b\) is used as a factor \(n\) times. A deep understanding of exponents is crucial for solving equations like those in the exercise, where expressions like \(2^n - 1\) determine sequences of numbers.

• Exponents follow specific rules, such as the power rule, product rule, and quotient rule.
• They are pivotal in scientific notation, helping express very large or small numbers succinctly.
• Exponential functions have widespread applications in growth and decay problems, finance, and natural sciences.

In the exercise, exponents play a central role in generating odd numbers. By systematically increasing the exponent \(n\), the expression \(2^n - 1\) provides sequential odd numbers, demonstrating the predictable nature of exponential growth in this context.