2(k+1) is just the next sequence.

You have:

2 4 6 8 10 12 14 ...

Essentially you're doing 2*1, 2*2, 2*3, 2*4 or...

2(1) 2(2) 2(3) 2(4) 2(5) 2(6) 2(7) ...

Man, that was tiring and I'm really lazy. Lets just use a variable ok?

2(k)

Boom, that was easy. I want the 4th sequence?

2(4) = 8

2 4 6 8 10 12 14 ...

See, it works. Thats the 4th one! (not really a proof, just go with it)

So... i'm lazy. Thus, to save myself some work, i'll just list a few, use these dots "..." and a variable notation to represent the rest.

2 4 6 ... 2(k)

How do I represent the sequence after the current one? Wait... what does that mean?!

Say we're on the 4th sequence (blue), how do we repesent the next one (red)

2 4 6 8 10 12 14 ...

Well, remember that we're just saying 2*1, 2*2, 2*3, thus:

2(1) 2(2) 2(3) 2(4) 2(5) 2(6) 2(7) ...

So we are on the 4th sequence and want to know whats next, well its the 5th sequence. The sequence after that is the 6th then 7th. How do we represent the 5th sequence and on in relationship to our current sequence... why don't we just +1? So 4 = 4th sequence, our current sequence. 4 + 1 = 5th sequence. 4 + 2 = 6th and so on.

2(1) 2(2) 2(3) 2(4) 2(4+1) 2(4+2) 2(4+3) ...

Thus if our current sequence is the 4th one, the next sequence is 4+1. The next next sequence is 4+1+1 or 4+2. The next next next sequence is 4+1+1+1 or 4+3 and so on.

Anyway, I only want to the next sequence, NOT the next next sequence and whatever. And listing everything out is kinda long. Let me just list a few, use k for the rest (current sequence) and k+1 for the next sequence.

2(1) 2(2) 2(3) 2(k) 2(k+1)

Lets test that. We're on the 4th sequence and the next sequence is the 5th.

2(4) 2(4+1)

8 10

Thus k+1 is just the next sequence.

With that out of the way, you're assuming the following statement is true.

2+4+6+...+2(k) = k^{2}+ k

So the next sequence, k+1, must also be true if the above statement is true:

2+4+6+...+2(k)+2(k+1) = k^{2}+ k + 2(k+1)

Where did the 2(k+1) come from? Its just the next sequence that we added on the left side.

**side note** I'm more used to doing a substituting k+1 for k on the right side but thats probably wrong haha. It ends up coming out to the same thing anyway.

Anyway:

k^{2}+ k + 2(k+1)

k^{2}+ k + 2k + 2

k^{2}+ k + 2k + 1 + 1

k^{2}+ 2k + 1 + k + 1

(k^{2}+ 2k + 1) + (k + 1)

(k+1)^{2}+ (k+1)

OR from your calculations:

k^{2}+3k + 2

k^{2}+ 2k + k + 1 + 1

k^{2}+ 2k + 1 + k + 1

(k^{2}+ 2k + 1) + (k + 1)

(k+1)^{2}+ (k+1)

The left side should technically be...

2 + 4 + 6 + ... + 2(k) + 2(k+1)

From our indictive hypothesis, 2 + 4 + 6 + ... + 2(k) = k^{2}+ k thus

2 + 4 + 6 + ... + 2(k) + 2(k+1) becomes

(k^{2}+k) + 2(k + 1)

Well, we just proved that didn't we?

Hopefully that makes sense. I'm trying to sound helpful and smart but all I really did was look at wikipedia LOL. You probably already figured this out already since this post is 3 or 4 days old =P.

Feel free to correct me.