Proof by Induction: Solving Algebraic Exercises

In summary, the conversation is about the topic of proof by induction and the speaker has attempted to solve a problem using this method. They are seeking feedback on their solution and guidance if they have made any mistakes. One person points out an error in the quoted line and explains how to correct it using the induction hypothesis. They also mention that the conversation should have been posted in the Homework & Coursework section.
  • #1
woundedtiger4
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Hi everyone,
I am trying to learn proof by induction method from http://en.m.wikibooks.org/wiki/Algebra/Proofs/Exercises
And I have tried to solve the second problem attached with this post. It will be great if someone can tell me if I am wrong anywhere and then guide me.

Thanks in advance.
 

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  • #2
woundedtiger4 said:
Hi everyone,
I am trying to learn proof by induction method from http://en.m.wikibooks.org/wiki/Algebra/Proofs/Exercises
And I have tried to solve the second problem attached with this post. It will be great if someone can tell me if I am wrong anywhere and then guide me.

Thanks in advance.
The line quoted below is wrong.
"It follows that (k + 1)! > 2k + 1"
This is precisely what you need to show! You can't just wave your arms and say that "it follows that ..." without showing it.

To show this, note that (k + 1)! = (k + 1)k!. Use your induction hypothesis (i.e., that k! > 2k) to finish the proof.

BTW, your thread should have been posted in the Homework & Coursework section, not in the technical math sections (especially not in the Linear and Abstract Algebra section. I am moving your thread.
 
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FAQ: Proof by Induction: Solving Algebraic Exercises

What is proof by induction?

Proof by induction is a mathematical technique used to prove that a statement is true for all natural numbers. It involves showing that the statement is true for the first natural number, and then using that assumption to prove that it is true for the next natural number. This process is repeated until it can be shown to be true for all natural numbers.

When is proof by induction used?

Proof by induction is typically used when trying to prove a statement that is true for all natural numbers, such as a formula or equation. It is also commonly used in discrete mathematics and computer science to prove properties of recursive algorithms.

What are the steps to solving an algebraic exercise using proof by induction?

The steps to solving an algebraic exercise using proof by induction are as follows:

  1. Step 1: Prove the base case (usually n=1) by substituting the value into the equation and showing that it is true.
  2. Step 2: Assume that the statement is true for some arbitrary natural number k.
  3. Step 3: Use this assumption to prove that the statement is also true for the next natural number (k+1).
  4. Step 4: Conclude that the statement is true for all natural numbers by showing that it is true for n=k+1.

What are some common mistakes to avoid when using proof by induction?

Common mistakes to avoid when using proof by induction include:

  • Forgetting to prove the base case.
  • Assuming that the statement is true for all natural numbers, rather than just for n=k+1.
  • Using circular reasoning by assuming the statement is true in the proof.
  • Not clearly stating the inductive hypothesis and how it is used in the proof.

Can proof by induction be used to prove statements for all real numbers?

No, proof by induction can only be used to prove statements for natural numbers. This is because the process relies on the fact that there is a "next" natural number after every number, which is not the case for real numbers. However, it is possible to use a similar technique called "complete induction" to prove statements for all real numbers.

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