Help with Mathematical Induction

In summary, the conversation discusses how to prove by mathematical induction that a specific expression will always be divisible by 11 for all positive integers. The conversation suggests starting by establishing a base case and then showing that if the proposition is true for n = k, it is also true for n = k+1. The use of modular arithmetic is also suggested as a method to simplify the problem.
  • #1
SeattleScoute
2
0

Homework Statement



Prove by matematical induction that (2^(n+1)+9(13^n)) divides by by 11 for all positive intergers


Homework Equations





The Attempt at a Solution



I really have no idea where to start...
 
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  • #2
The problem outright tells you a place to start!
 
  • #3
Is P1 true? If so, then if I say Pk is true is Pk+1 also true?
 
  • #4
jegues said:
Is P1 true? If so, then if I say Pk is true is Pk+1 also true?
Are you also SeattleScoute?

That's basically what is needed. Establish a base case. Assume the proposition is true for n = k. Show that P(k) being true implies that P(k+1) is also true.
 
  • #5
Are you also SeattleScoute?

No I'm not, I thought I'd help :S
 
  • #6
you may use modular arithmetic to lighten your job .
 

FAQ: Help with Mathematical Induction

1. What is mathematical induction and how does it work?

Mathematical induction is a proof technique used to show that a statement is true for all natural numbers. It works by first proving that the statement is true for the base case (usually n=1), and then assuming that the statement is true for some arbitrary number n and using that assumption to prove that it is also true for n+1. This process is repeated until the statement is shown to be true for all natural numbers.

2. Why is mathematical induction important in mathematics?

Mathematical induction is important because it allows us to prove statements that hold true for all natural numbers without having to check each individual case. This can save a lot of time and effort when dealing with complex mathematical problems.

3. What are the steps involved in a mathematical induction proof?

The steps involved in a mathematical induction proof are:

  • Prove the statement is true for the base case
  • Assume the statement is true for an arbitrary number n
  • Use the assumption to prove that the statement is also true for n+1
  • Repeat this process until the statement is shown to be true for all natural numbers

4. Can mathematical induction be used to prove any statement?

No, mathematical induction can only be used to prove statements that hold true for all natural numbers. It cannot be used to prove statements about real numbers or other types of mathematical objects.

5. What are some common mistakes to avoid when using mathematical induction?

Some common mistakes to avoid when using mathematical induction include:

  • Not proving the base case
  • Assuming the statement is true for n+1 without first proving it for n
  • Using circular reasoning by assuming the statement is true in the proof
  • Not using the correct induction hypothesis

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