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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