Prove using the principle of mathematical induction: (i) the number of diagonals of a convex polygon with n vertices is n(n − 3)/2, for n ≥ 4, (ii) 2n < n! for for all n > k > 0, discover the value of k before doing induction.

Step-by-step explanation:

Proof for i)

We will prove by mathematical induction that, for every natural , the number of diagonals of a convex polygon with n vertices is .

In this proof we will use the expression d(n) to denote the number of diagonals of a convex polygon with n vertices

Base case:

First, observe that:, for n=4, the number of diagonals is

Inductive hypothesis:  

Given a natural ,

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that, given a convex polygon with n vertices, wich we will denote by P(n), if we add a new vertix (transforming P(n) into a convex polygon with n+1 vertices, wich we will denote by P(n+1)) we have that:

Because of these observation we know that, for every ,

Therefore:

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural ,

Proof for ii)

Observe that:

Then, the statement is not true for n=1,2,3.

We will prove by mathematical induction that, for every natural ,

.

Base case:

For n=4,

Inductive hypothesis:  

Given a natural ,

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that,

wich is true as we are assuming . Therefore:

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural ,

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