# Math Induction Explained: Step-by-Step Guide

• Mathematica
• Richter915
In summary: Unless the P(0)'th term is 0, it doesn't prove much of anything to show that P(1) is greater than zero (Consider an increasing sequence that starts with negative numbers). This inductive proof starts with showing P(2)>P(1).
Richter915
Hi I was wondering if any of you could explain the concept of Mathematical Induction to me...I really am confused on this, we didn't learn it in class (yet we're going to be quizzed on it)...if you could walk through a simple example, it would be really appreciated...thank you!

Wow, thank you so much. I have a question though. My teacher, as an example asked to use induction to prove that a certain sequence is increasing...so by using induction would I:
1. Prove P(1) exists
2. Prove P(K) is increasing
3. Prove P(K+1) > P(K) thus concluding that the sequence increases>

Richter915 said:
Wow, thank you so much. I have a question though. My teacher, as an example asked to use induction to prove that a certain sequence is increasing...so by using induction would I:
1. Prove P(1) exists
2. Prove P(K) is increasing
3. Prove P(K+1) > P(K) thus concluding that the sequence increases>
The first step is not to prove that P(1) exists, it is to prove that the first theorem in a chain of theorems holds true. The "arbitrary kth theorem being true implies truth of (k+1)th theorem" proof then allows one to form a logical chain to the (k+1)-th theorem from the 1st theorem if challenged to prove that the (k+1)-th theorem is true.
In your case, you want to prove that the second item in the sequence is greater than the first item. Your second step is to prove that given an arbitrary kth term in the sequence and assuming the kth term is greater than the (k-1)th term, prove that the (k+1)th term is greater than or equal to (if your aim is not to prove strictly increasing) the kth term. The actual item to prove changes with each set of theorems, only the logical machinery is the same.

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oh ok I see my mistake...so I'd have to say that P(1) holds true for the condition...so for this example it'd be P(1)>0 and then I'd proceed from there?

Richter915 said:
oh ok I see my mistake...so I'd have to say that P(1) holds true for the condition...so for this example it'd be P(1)>0 and then I'd proceed from there?
Unless the P(0)'th term is 0, it doesn't prove much of anything to show that P(1) is greater than zero (Consider an increasing sequence that starts with negative numbers). This inductive proof starts with showing P(2)>P(1).

## 1. What is mathematical induction?

Mathematical induction is a proof technique used in mathematics to show that a statement or property holds for all natural numbers. It involves proving that the statement is true for the first natural number (usually 1), and then showing that if it is true for any given natural number, it is also true for the next natural number.

## 2. How does mathematical induction work?

Mathematical induction works by breaking down a statement or property into smaller parts and proving that each part holds for a given natural number. This is done in two steps: the base case, where the statement is proven to be true for the first natural number, and the inductive step, where it is shown that if the statement holds for any given natural number, it also holds for the next natural number.

## 3. When is mathematical induction used?

Mathematical induction is used when proving statements or properties that involve natural numbers, such as inequalities, divisibility, and series. It is also used in computer science and logic to prove the correctness of algorithms and statements.

## 4. What are the advantages of using mathematical induction?

One advantage of using mathematical induction is that it provides a systematic and rigorous way of proving statements that hold for all natural numbers. It also allows for the proof of more complex statements by breaking them down into smaller parts. In addition, mathematical induction is widely applicable and can be used in various areas of mathematics and computer science.

## 5. Are there any limitations to mathematical induction?

While mathematical induction is a powerful proof technique, it does have some limitations. It can only be used to prove statements that hold for all natural numbers, and it may not be applicable to all types of mathematical problems. Additionally, the inductive step may be difficult to prove in some cases, making the overall proof more challenging.

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