4+8+12+..... 4n = 2n(n+1). Using principle of mathematical induction , prove the following for all n E N ?

 First we verify that it is true for 1, i.e. 4n does equal 2n(n + 1) as 4*1 = 2*1*(1 + 1) as 4 = 2*2. 

Now, assume that it holds for some integer k. 
i.e. That 4 + 8 + 12 + ... + 4k = 2k(k + 1) 

We now wish to show that it is also true for k + 1. 
That is, we need to prove that if it is true for k, then 
4 + 8 + 12 + ... + 4k + 4(k + 1) = 2(k + 1)(k + 1 + 1) 
or 4 + 8 + 12 + ... + 4k + 4(k + 1) = 2(k + 1)(k + 2) 

So, from the inductive hypothesis we have 
4 + 8 + 12 + ... + 4k + 4(k + 1) 
= 2k(k + 1) + 4(k + 1) 
k + 1 is a common factor, so we have that is equal to 
(2k + 4)(k + 1) 
= 2(k + 2)(k + 1) 
= 2(k + 1)(k + 2) as we wished to show. 

So, from the principle of induction, it is true for all natural numbers.