Answer:
Explanation:
First, observe that:

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

We will prove our base case (when n=1) to be true.
Base case:

Inductive hypothesis:
Given a natural n,

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

Then, by the observation made at the beginning of this proof, we have that

With this we have proved our statement to be true for n+1.
In conlusion, for every natural n
,

b)
Observe that
Then,
- If
- If
- If
Therefore, for every
c)
We will prove by mathematical induction that, for every natural n>3,
Inductive hypothesis:
Given a natural n>4,
[tex](n-1)^2<2n^2.[tex]
Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.
Inductive step:
[tex]((n+1)-1)^2=((n-1)+1)^2=(n-1)^2+2(n-1)+1<2n^2+2(n-1)+1=2n^2+2n- 1<2n^2+2n+1 =2(n+1)^2.[tex]
With this we have proved our statement to be true for n+1.
In conclusion, for every natural n>3,
[tex](n-1)^2<2n^2.[tex]