How to prove by mathematical induction?

In summary, mathematical induction is a proof technique used to prove statements about all positive integers. It works by showing that the statement is true for a base case, and then assuming it is true for some value n and proving that it is also true for n+1. This can be illustrated with the example of proving an equation using induction, where the base case is shown to be true and then the equation is proven to hold for n and n+1. This technique can be confusing at first, but with practice and understanding of the concept, it can be a useful tool in mathematical proofs.
  • #1
James2
35
0
How do I prove a formula/rule or something by mathematical induction? Please give me a few examples or resources and explain it as best you can because I think I'm messing up some how.
 
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  • #2
What are you not getting, exactly? If you just don't know what induction is, surely a google search would be faster than starting a new thread.
 
  • #3
Everything I read confuses me, it tells me to do something different everytime...
 
  • #4
James2 said:
Everything I read confuses me, it tells me to do something different everytime...

We can't help you if you don't explain what's confusing you. Try posting your attempt at solving an induction problem and explain where you get stuck.
 
  • #5
I have an equation, (5n + 2) = 2[(5/2)n + 1] I know this is true from the basis step. Then I asume n = k now I must prove n = k + 1. So, (5k + 2) = 2[(5/2)k + 1]

Alright then, I try to substitute k + 1 in and add it or something so I get... 2[(5/2)k + 1] + [5(k + 1) + 2] = 2 [(5/2)(k + 1) + 1]

Simplifying, I get 10k + 2 + 5k + 5 + 2 = 10(k + 1) + 2

And finallly, 15k + 9 =/= 10k + 12

SO... whaaaat? What happened here?
 
  • #6
(5n + 2) = 2[(5/2)n + 1]

n = 0:
5*0 + 2 = 2[(5/2)*0 + 1]

The case is true for 0.

Suppose the case is true for n = k.
Now we can use (5k + 2) = 2[(5/2)k + 1].

n = k + 1:
5(k + 1) + 2 = (5k + 2) + 5 = 2[(5/2)k + 1] + 5 = 2[(5/2)k + 1 + 5/2] = 2[(5/2)(k+1) + 1]

The case n = k + 1 follows from the case n = k.
With case n = 0 true the equation therefore works for all non-negative integers.
 

1. Can you explain the concept of mathematical induction?

Mathematical induction is a method of proving a statement or theorem in mathematics. It involves breaking down a problem into smaller, simpler cases and proving that the statement holds for each case. This method is based on the principle that if a statement is true for a particular case, it can be assumed to be true for the next case as well.

2. How do I know when to use mathematical induction?

Mathematical induction is typically used to prove statements involving natural numbers or integers. It is most commonly used to prove statements about sequences, series, and divisibility. If you are trying to prove a statement that involves these concepts, then mathematical induction may be a suitable method to use.

3. What are the steps for using mathematical induction to prove a statement?

The steps for using mathematical induction are as follows:

  1. Base case: Prove that the statement holds for the first case, usually when n = 1.
  2. Inductive hypothesis: Assume that the statement holds for a specific case, usually when n = k.
  3. Inductive step: Use the inductive hypothesis to prove that the statement holds for the next case, when n = k+1.
  4. Conclusion: By the principle of mathematical induction, the statement holds for all cases.

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

One common mistake when using mathematical induction is assuming that the statement holds for all cases without properly proving it for each case. It is important to remember that the inductive step must be proven for each case in order for the statement to be considered true for all cases. Another mistake is using incorrect or incomplete notation, which can lead to incorrect proofs.

5. Are there any alternative methods to mathematical induction for proving statements?

Yes, there are other methods for proving statements in mathematics, such as direct proof, proof by contradiction, and proof by contrapositive. These methods may be more suitable depending on the statement and the problem at hand. It is important to understand and be familiar with multiple proof techniques in order to effectively solve mathematical problems.

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