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Chapter 11: Problem 12

Use mathematical induction to prove each of the following. $$3^{n}<3^{n+1}$$

Short Answer

Expert verified
By mathematical induction, \(3^n < 3^{n+1}\) is true for all non-negative integers n.

Step by step solution

01

Base Case

Check the base case for the smallest value of n. Let n = 0: We need to show that the inequality holds for n = 0. Substituting n = 0:\(3^0 < 3^{0+1}\)Simplifies to:\(1 < 3\)Since 1 is less than 3, the base case holds true.
02

Inductive Hypothesis

Assume the inequality is true for some arbitrary positive integer k. That is, assume:\(3^k < 3^{k+1}\)This assumption is called the inductive hypothesis.
03

Inductive Step

We need to show that the inequality also holds for k + 1, that is, we need to show:\(3^{k+1} < 3^{k+2}\)Using the inductive hypothesis \(3^k < 3^{k+1}\):Multiply both sides of the hypothesis by 3:\(3 \times 3^k < 3 \times 3^{k+1}\)Simplifies to:\(3^{k+1} < 3^{k+2}\)Thus, the inequality holds for k + 1.
04

Conclusion

Since we have shown the base case is true and the inequality holds for k + 1 if it holds for k (inductive step), by the principle of mathematical induction, the inequality \(3^n < 3^{n+1}\) is true for all non-negative integers n.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base Case
In mathematical induction, the base case is the first step where we verify the statement for the smallest value of the variable, usually denoted as n. It lays the groundwork for the rest of the proof. Let's consider our example:

Let n = 0.

We substitute n into our inequality to see if it holds true: \(3^0 < 3^{0+1}\).

This simplifies to \(1 < 3\), which is clearly true. Therefore, the base case holds.

Getting the base case right is crucial because it proves the initial step from where our inductive proof can begin.
Inductive Hypothesis
The inductive hypothesis is the second step in a mathematical induction proof. In this step, you assume the statement is true for some arbitrary positive integer k.

For our inequality, we assume:

\(3^k < 3^{k+1}\).

This is not a proof but an assumption that helps in the next step. The assumption is vital because it allows us to build on an assumed truth to show that if the statement holds for one integer, it holds for the next.
Inductive Step
The inductive step is where we show that if the statement is true for an arbitrary integer k (our inductive hypothesis), then it must also be true for k + 1.

From our hypothesis, \(3^k < 3^{k+1}\), we need to prove that:

  • \(3^{k+1} < 3^{k+2}\).


Using the inductive hypothesis, we multiply both sides of \(3^k < 3^{k+1}\) by 3:

  • \(3 \times 3^k < 3 \times 3^{k+1}\)
  • \(3^{k+1} < 3^{k+2}\)


We've shown that if the inequality holds for k, it also holds for k+1.
Inequality Proofs
Proving inequalities using mathematical induction follows a systematic process similar to proving equalities. You begin with the base case, assume for some arbitrary k, and then prove for k + 1.

In our example, we successfully showed that:
  • \(3^n < 3^{n+1}\)


    • is true for all non-negative integers n.

    Effective inequality proofs often involve steps such as:
    • Verification of the base case to establish a starting point.
    • Assuming the statement holds for an arbitrary value.
    • Using the assumption to prove the next value in the sequence.


    These steps ensure that the inequality holds universally across the specified range of numbers.

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