Use mathematical induction to prove that the formula
1^3+3^3+5^3+-1)^3=n^2(2n^2-1) is true for all natural n.
(b) show tha n^3-n+3 is divisible by 3 for all natural numbers n.
(c) usw mathematical induction to prove that (n-1)^2< 2n^2 for all natural numbwrs n> 3

Step-by-step 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,

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]