A student claims that 2 + 4 + ...+ 2n = (n+ 2)(n - 1) for any integer n =2 and presents the following proof.

We assume that P(n) : 2 + 4 + ... + 2n = (n+ 2)(n -1), and we want to prove
P(n + 1) : 2 + 4 + ...+ 2n + 2(n + 1) = (n + 3)n.
2 + 4 + ...+ 2n + 2(n + 1)
= 2 + 4 + ... + 2n+ 2 n + 2
= (n+ 2)(n - 1) + 2n + 2
=n(n+ 2) - (n+ 2) + 2n + 2
=n(n + 2) +n
=n(n+ 3)
P(n+1) is true, hence the claim is true

Is there anything wrong with this proof?

Question

A student claims that 2 + 4 + ...+ 2n = (n+ 2)(n - 1) for any integer n >=2 and presents the following proof.

We assume that P(n) : 2 + 4 + ... + 2n = (n+ 2)(n -1), and we want to prove 
P(n + 1) : 2 + 4 + ...+ 2n + 2(n + 1) = (n + 3)n.
2 + 4 + ...+ 2n + 2(n + 1) 
= 2 + 4 + ... + 2n+ 2 n + 2
= (n+ 2)(n - 1) + 2n + 2
=n(n+ 2) - (n+ 2) + 2n + 2
=n(n + 2) +n
=n(n+ 3)
P(n+1) is true, hence the claim is true

Is there anything wrong with this proof?

anonymous · Answer

I think what is wrong here is at the very start, 2(n+1) should just be 2n + 1. Then every other step looks good except obviously the claim is false?

anonymous · Answer

If you are using induction, your proof is incomplete.

anonymous · Answer

Do you want to know why?

anonymous · Answer

It didn't prove the base case.