Display a proof for the following theorems:

Question

Display a proof for the following theorems:
3 divides n³ +2n for positive integers n.

Explanation

In the question, we have to Display proof for the theory.

So now we have already given a theorem and that theorem is 3 divides n³ +2n for positive integers n.

And we will prove this theorem with the end end result. So we will use the Principal of Mathematical Induction.

k=0, U(k)=0

So now we can see that 0 is divisible by 3, so U(0) will be true here.

So now, we will again check this. But with the different value of k. And the value of k this time will be 1.

U(1) = 1+2 = 3

And now here also 3 is divisible by 3, so we have U(1)  also true here.

And now we have to consider. So Consider or assume that the end result for k-q is always true. That means if we put any number in the place of k will always give us the true output. But we will always put the positive integer.

U(q)= is divisible by 3.
U(q)=3m —(i)   This is our equation 1

So next we have to prove that the end result is true for k=q+1
So, U(q+1) =
Now in the above equation if we use identity. And we get the following

U(q=1)=

Using (i)
= 3m + 3
=a number divisible by 3.
Thus, U(q+1) is also true.

So Hence after all this, we can say that. Using the Principle of Mathematical Induction, there are 3 divides n³ +2n for positive integers n.

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