Get a Kick on Induction Proof for n²<=n!

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In summary, The homework statement is asking for help with a problem in induction proof. The Attempt at a Solution says that the reasoning is as follows: you have checked by hand that it is true for n = 1. You have proven that if it is true for n0 = 1 (which it is), then it is true for n = n0 + 1 = 2. So it is true for n = 2. Also, you have shown that if it is true for n = 2, it is true for n = 3. Since it is true for n = 2, it holds for n = 3. Similarly, it is true for n = 4, and for n = 5, and so on. The problem is
  • #1
Kasperloeye
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Homework Statement



Hey I'm a little fuzzy on how the induction-proof really works.. and am therefore a little stuck.. I know some parts.. but I need a kick in the right direction.. you don't have to solve it.. just give me a hint or a kick as I said ^^ that would be nice

Homework Equations


Which nonnegative integers makes the following statement true.. a induction proof would be the best way of solving it i think.. but how do I get started.. and what would be the base theorem..?
n²<=n!


The Attempt at a Solution


I can logically see what values that is needed.. but help.. please..
 
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  • #2
Generally, when you have some statement like "for all n, X is true", a proof by induction consists of two steps. First, you have to show that for some simple case (usually n = 0 or 1, depending on the question), X is true. Then you assume that X is true for all integers n up to some given value n0, and you prove that under that assumption, X is also true for n0 + 1.

The reasoning is then as follows: you have checked by hand that it is true for n = 1. You have proven that if it is true for n0 = 1 (which it is), then it is true for n = n0 + 1 = 2. So it is true for n = 2. Also, you have shown that if it is true for n = 2, it is true for n = 3. Since it is true for n = 2, it holds for n = 3. Similarly, it is true for n = 4, and for n = 5, and so on.

Note that proof by induction is a convenient way (once you are used to it) to prove such statements "for all n", but it doesn't help you to find the statement. So if instead of "prove that the nth derivative is ...formula..." you get "derive and prove a formula for the nth derivative", you will first need to come up with a hypothesis by some other way. Once you have the hypothesis, you can make it into a theorem and try to prove it by induction.

In your particular problem, you could consider a statement like "For all n >= 2, n2 <= (n!)".
 
  • #3
CompuChip said:
In your particular problem, you could consider a statement like "For all n >= 2, n2 <= (n!)".

Hehe that's where it get's funny.. because n>= 2, n²<=n! is like 2²<=2! 4 <= 2 .. so it has to n /= 2


So Basis should be n=4 since

4²<=4!
16 <=24

SO: For n=k, k²<=k!

but if we take it with n= k+1 it's (k+1)² <=(k+1)!
(k+1)² <= (k+1)(k!)
(k+1) <= k!
but our hypothasis says k^2 <= k!, and because k+1 <= k^2 (for k>=4) then
k+1 <= k^2 <= k!.

right??
 

1. What is induction proof?

Induction proof is a mathematical method used to prove that a statement is true for all natural numbers. It involves proving that the statement is true for the first natural number, and then showing that if the statement is true for one natural number, it is also true for the next natural number.

2. How does induction proof work?

Induction proof works by establishing a base case (usually n=1) where the statement is true, and then using the assumption that the statement is true for n=k to prove that it is also true for n=k+1.

3. What is the statement being proved in "Get a Kick on Induction Proof for n²<=n!"?

The statement being proved is n²<=n, which means that the square of any natural number is less than or equal to the number itself.

4. Why is it important to use induction proof in mathematics?

Induction proof is important in mathematics because it allows us to prove statements that are true for all natural numbers without having to check each individual case. It also helps to establish patterns and generalize solutions.

5. Can induction proof be used to prove statements for other types of numbers?

Yes, induction proof can also be used to prove statements for other types of numbers such as integers, real numbers, and complex numbers. However, the base case and the method of proving the statement may differ for each type of number.

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