Prove that $n$ lines form $[n(n+1)/2]+1$

Question

I saw the same formula on the web as $(n^2+n+2)/2$. I saw the same problem here. However, I didn't understand very well. I have to prove it and it seems I should use induction. So I decided to do it "my way, which I can understand for myself": ( :-) )

1. I proved that with $n=1$, the formula works
2. I assume it works with $n=k => [k(k+1)/2] $should be equal to the number of planes, made by k lines.
3. But then I cannot reach the same formula for $k+1$

I have noticed however the following (let's say I add the$ (k+1)$th line): This line can be split by theother lines into $k+1$ parts and can therefore add $k+1$ more planes. Then the number of planes should be: $1+2+3+4+...+k+k+1$ but that equals $[(k+1)(k+2)/2].$ The $ +1$ is missing. What's wrong?

Thank you in advance!