Problem 16 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-26: Prove that the statement is true for every positive integer \(n\). $$ 1+2 n \leq 3^{n} $$

Step-by-Step Solution

Answer

By induction, the inequality \(1+2n \leq 3^n\) holds for all positive integers \(n\).

Step 1: Understand the Base Case

To start a mathematical induction proof, we verify the base case. For our hypothesis, the statement can be tested when \(n = 1\). Substitute \(n = 1\) into the inequality to see if it holds: \(1 + 2 \times 1 = 3\) and \(3^1 = 3\). Thus, \(1 + 2 \times 1 \leq 3^1\) holds true. So the base case is proved.

Step 2: State the Inductive Hypothesis

Assume that the inequality \(1 + 2n \leq 3^n\) is true for some positive integer \(k\). This means that \(1 + 2k \leq 3^k\). This assumption will be used to prove the statement is true for \(k + 1\).

Step 3: Prove the Inductive Step

We need to prove that if the statement holds for \(n = k\), then it also holds for \(n = k+1\). Start with the left-hand side of the inequality: \(1 + 2(k + 1) = 1 + 2k + 2\). By the inductive hypothesis, \(1 + 2k \leq 3^k\), so \(1 + 2k + 2 \leq 3^k + 2\).

Step 4: Simplify and Conclude

Next, we need to show \(3^k + 2 \leq 3^{k+1}\). Since \(3^k + 2 < 3 \times 3^k = 3^{k+1}\) for \(k \geq 1\), the inequality \(3^k + 2 \leq 3^{k+1}\) holds true. Thus, our statement \(1 + 2(k+1) \leq 3^{k+1}\) is verified, completing the induction step. Therefore, by the principle of induction, the original inequality holds for all positive integers \(n\).

Key Concepts

Base Case

The base case is where we kick off our mathematical induction. It's like setting a solid foundation for a building. In order to prove our statement by induction, we first need to verify it for the smallest positive integer, which is usually 1. For this exercise, the inequality is \(1 + 2n \leq 3^n\). We substitute \(n = 1\) into the equation.

• Calculate the left side: \(1 + 2 \times 1 = 3\).
• Calculate the right side: \(3^1 = 3\).
• Check: \(3 \leq 3\), which is true.

We've proven the base case, which sets the stage for the steps that follow. Without a valid base case, the entire induction process would fall apart.

Inductive Hypothesis

The inductive hypothesis can be seen as the assumption step, where we assume that our statement holds true for a particular case, say \(n = k\). This is a crucial part of induction because it's the stepping stone to proving a more general case.

• The hypothesis assumes: \(1 + 2k \leq 3^k\) holds for some integer \(k\).
• This assumption is temporary and will be tested in the next step, the inductive step.

Just like a scientific hypothesis, it sets up conditions for the next part of our proof. Keep in mind, the goal is to use this hypothesis to validate the statement \(n = k + 1\).

Inductive Step

In the inductive step, we work on proving that if the inductive hypothesis holds true for \(n = k\), it should also be true for \(n = k + 1\). Basically, we're taking a small step forward by building on our assumption.- Start with the left side of the inequality for \(n = k+1\): - Calculate: \(1 + 2(k + 1) = 1 + 2k + 2\). - Using our inductive hypothesis: Since \(1 + 2k \leq 3^k\), we rewrite it as \(1 + 2k + 2 \leq 3^k + 2\).- Now, we need to ensure that: - \(3^k + 2 \leq 3^{k+1}\), which simplifies our task of extending the truth to the next case.This completes the inductive step. Each step is carefully constructed, using previous truths to establish the next logical conclusion.

Inequality Proof

This exercise wraps up with an inequality proof. After confirming our base case and extending our hypothesis to the next integer, we need to ensure the inequality is correct throughout.- We justified that \(3^k + 2 < 3 \times 3^k = 3^{k+1}\). - This means that starting from a truthful base, we've shown the inequality \(1 + 2(k+1) \leq 3^{k+1}\) holds by induction.- Proving inequalities often involves: - Simplifying expressions, as seen when comparing \(3^k + 2\) and \(3^{k+1}\). - Logical reasoning to ground each conclusion.By completing the inductive step, we cement the validity of our statement for all positive integers \(n\). Thus, induction serves as a powerful tool for establishing inequality proofs like this one.