13. What Is The Second Step In Proving By Mathematical Induction That For Every Positive Integer \[$ N \$\], \[$ 11^n - 6 \$\] Is Divisible By 5?A. Let \[$ N = K \$\]. Then Assume That \[$ 11^{k+1} - 6 \$\] Is Divisible

Introduction

Mathematical induction is a powerful technique used to prove that a statement is true for all positive integers. It involves two main steps: the base case and the inductive step. In this article, we will focus on the second step in proving by mathematical induction that for every positive integer $n$, $11^n - 6$ is divisible by 5.

Understanding the Problem

The problem requires us to prove that for every positive integer $n$, the expression $11^n - 6$ is divisible by 5. This means that we need to show that $11^n - 6$ can be expressed as a multiple of 5, i.e., $11^n - 6 = 5m$, where $m$ is an integer.

The Second Step in Mathematical Induction

The second step in mathematical induction is to assume that the statement is true for some positive integer $k$, and then show that it is true for $k+1$. In other words, we assume that $11^k - 6$ is divisible by 5, and then show that $11^{k+1} - 6$ is also divisible by 5.

Assuming the Inductive Hypothesis

Let $n = k$. Then assume that $11^{k+1} - 6$ is divisible by 5. This is our inductive hypothesis, which we will use to prove that the statement is true for $k+1$.

Proving the Inductive Step

To prove the inductive step, we need to show that if $11^{k+1} - 6$ is divisible by 5, then $11^{k+2} - 6$ is also divisible by 5. We can do this by using the inductive hypothesis and some algebraic manipulations.

Expanding the Expression

We start by expanding the expression $11^{k+2} - 6$:

$$11^{k+2} - 6 = 11^{k+1} \cdot 11 - 6$$

Using the Inductive Hypothesis

We can use the inductive hypothesis to substitute $11^{k+1} - 6$ with $5m$, where $m$ is an integer:

$$11^{k+2} - 6 = 11^{k+1} \cdot 11 - 6 = (5m) \cdot 11 - 6$$

Simplifying the Expression

We can simplify the expression further by multiplying $5m$ by 11:

$$11^{k+2} - 6 = 55m - 6$$

Factoring Out 5

We can factor out 5 from the expression:

$$11^{k+2} - 6 = 5(11m - 1)$$

Conclusion

We have shown that if $11^{k+1} - 6$ is divisible by 5, then $11^{k+2} - 6$ is also divisible by 5. This completes the inductive step, and we have successfully proved that for every positive integer $n$, $11^n - 6$ is divisible by 5.

Conclusion

In this article, we have discussed the second step in proving by mathematical induction that for every positive integer $n$, $11^n - 6$ is divisible by 5. We have assumed the inductive hypothesis, expanded the expression, used the inductive hypothesis, simplified the expression, and factored out 5 to show that the statement is true for $k+1$. This completes the inductive step, and we have successfully proved the statement.

Final Thoughts

$$$$$$

Introduction

In our previous article, we discussed the second step in proving by mathematical induction that for every positive integer $n$, $11^n - 6$ is divisible by 5. In this article, we will provide a Q&A guide to help you understand the concept of mathematical induction and how to apply it to prove a statement.

Q: What is mathematical induction?

A: Mathematical induction is a technique used to prove that a statement is true for all positive integers. It involves two main steps: the base case and the inductive step.

Q: What is the base case in mathematical induction?

A: The base case is the first step in mathematical induction, where we prove that the statement is true for the smallest positive integer, usually 1.

Q: What is the inductive step in mathematical induction?

A: The inductive step is the second step in mathematical induction, where we assume that the statement is true for some positive integer $k$, and then show that it is true for $k+1$.

Q: How do I prove the inductive step?

A: To prove the inductive step, you need to assume the inductive hypothesis, expand the expression, use the inductive hypothesis, simplify the expression, and factor out any common factors.

Q: What is the inductive hypothesis?

A: The inductive hypothesis is the assumption that the statement is true for some positive integer $k$. This is used to prove that the statement is true for $k+1$.

Q: How do I use the inductive hypothesis?

A: You can use the inductive hypothesis by substituting the expression $11^{k+1} - 6$ with $5m$, where $m$ is an integer.

Q: What is the difference between the base case and the inductive step?

A: The base case is the first step in mathematical induction, where we prove that the statement is true for the smallest positive integer. The inductive step is the second step, where we assume that the statement is true for some positive integer $k$, and then show that it is true for $k+1$.

Q: Can I use mathematical induction to prove any statement?

A: No, you cannot use mathematical induction to prove any statement. Mathematical induction is only applicable to statements that involve a sequence of positive integers.

Q: What are some common mistakes to avoid when using mathematical induction?

A: Some common mistakes to avoid when using mathematical induction include:

• Not assuming the inductive hypothesis
• Not expanding the expression correctly
• Not using the inductive hypothesis correctly
• Not simplifying the expression correctly
• Not factoring out any common factors

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of mathematical induction and how to apply it to prove a statement. We have discussed the base case, the inductive step, the inductive hypothesis, and how to use the inductive hypothesis. We have also provided some common mistakes to avoid when using mathematical induction.

Final Thoughts

Mathematical induction is a powerful technique used to prove that a statement is true for all positive integers. By understanding the concept of mathematical induction and how to apply it, you can prove a wide range of statements. Remember to assume the inductive hypothesis, expand the expression, use the inductive hypothesis, simplify the expression, and factor out any common factors to prove the inductive step.