Prove that for all natural numbers n, n+3 < 5(n^2)

Once again we will use induction to solve this proof.

First we prove the case to be true for the first natural number:1

1+3 < 5*(1^2) that is to say 4 < 5

Now we will try prove the case to be true for n+1 to do this, we will add one to both sides and get:

(n+1) + 3 < 5(n^2) + 1

5(n^2) + 1 < 5 = 5(n+1)^2

thus

(n+1) + 3 < 5(n+1) ^2

The reason why we did all that algebra in the middle was to get the form n+1 on both sides which is important to use the principle of mathematical induction to conclude that n+3<5(n^2) for all n in the natural numbers. ~~~~

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