Question

Prove the following statements with either induction, strong induction or proof by smallest counterexample. Prove that \(2^{n}+1 \leq 3^{n}\) for every positive integer \(n\).

Short answer

The inequality \(2^{n}+1 \leq 3^{n}\) is true for all positive integers \(n\). This is confirmed by mathematical proof using the method of mathematical induction.

Step-by-step solution

Base Case

We need to check if the statement holds for the lowest positive integer, which is \(n = 1\). Substituting into the statement gives us \(2^{1}+1 \leq 3^{1}\) which simplifies to 3 ≤ 3, and this is true.

Assume for a General Case

Assume the statement is true for some positive integer \(n = k\). This means that \(2^{k}+1 \leq 3^{k}\). This is the Inductive Hypothesis.

Proof for the Next Case

Now we want to prove that if the statement is true for \(n = k\), then it will also be true for \(n = k+1\). So, we need to prove that \(2^{k+1}+1 \leq 3^{k+1}\). We can start from the Inductive Hypothesis \(2^{k}+1 \leq 3^{k}\), multiply both sides of the inequality by 2 (keeping in mind that the inequality does not change as 2 is positive), then make use of the fact that 2*2^{k} = 2^{k+1} and 2*3^{k} = 2*3^{k+1}/3. The inequality becomes \(2^{k+1} + 2 \leq 2*3^{k+1}/3\). Since this is greater than \(2^{k+1} + 1\), and it's also less than \(3^{k+1}\) because \(2*3^{k+1}/3\) is two thirds of \(3^{k+1}\), this completes the induction.

Key concepts

Proof by Induction

Mathematical induction is a powerful proof technique often used to establish the validity of statements pertaining to natural numbers. It mirrors the domino effect, where knocking down the first domino causes a chain reaction that knocks down an entire sequence of dominos.

Induction has two crucial steps: the base case and the inductive step. The base case checks if the statement holds for the initial number in the sequence, usually 1. In our exercise, we started with checking that the inequality holds for the lowest positive integer, which is 1. Confirming this step ensures that our 'first domino' stands correctly.

Next, we move to the assumption, or the inductive hypothesis, where we assume that the given statement is true for some arbitrary positive integer 'k'. This assumption is like saying, 'If one domino falls, we suppose the next one will too.' Finally, we prove the statement for 'k+1', the next domino in the sequence. By proving that the true statement for 'k' implies it is also true for 'k+1', we effectively set off the domino chain reaction, substantiating the veracity of the statement for all subsequent numbers.

Inequalities

Inequalities express the relationship between two values when they are not equivalent. They are a foundational concept in algebra and are used throughout mathematics to describe order and to indicate that one quantity is less than, greater than, less than or equal to, or greater than or equal to another quantity.

Handling inequalities requires care, particularly with operations that can affect the direction of the inequality, such as multiplication or division by a negative number. An important feature of inequalities is that you can add or subtract the same value on both sides without changing the inequality's direction, which is part of the standard process in solving them.

The exercise we are considering involves an inequality with exponential functions. When we multiplied the expression by 2, the direction of the inequality did not change because 2 is positive. We continued by comparing exponential terms, which is a common strategy when dealing with inequalities that involve exponential expressions.

Exponential Functions

Exponential functions involve an exponent that is a variable; they are expressed in the form of f(x) = a^x, where 'a' is a positive real number not equal to 1. These functions model a variety of real-world phenomena and are characterized by rapid growth or decay. Different bases of exponential functions grow at different rates—this property is fundamental when comparing different exponential terms in inequalities.

In the given exercise, we compare the growth rates of two exponential functions, namely, 2^n and 3^n. Recognizing that 3^n grows faster than 2^n is key to understanding why the inequality 2^n + 1 ≤ 3^n holds true for all positive integers. The proof by induction bolsters this comparison by proving that at any given step 'k', moving to 'k+1' does not disrupt the established inequality, thus confirming that the faster growth rate is consistently maintained.