This might be a silly question but I have never had proofs at school and I get stuck at the uni especially in a foreign country. So here goes...
The following needs to be proven by Mathematical Induction:
This is an example in my Script so here is the solution as well:
Step 1: (I will skip this, it's just n=1 and it is correct)
Step 2: $A(n) \implies A(n+1)$
Well, I understand what is written there, but, when I 'follow the instructions' of the Mathematical Proof I get something like this (I basically put $n+1$ instead of $n$):
$$\sum_{k=1}^{n+1} k = \frac {(n+1)*(n+2)}{2} = (n+1) + \sum_{k=1}^{n} k = \frac {(n+1)*(n+2)}{2}$$
$$\sum_{k=1}^{n} k = \frac {(n+1)*(n+2)}{2} - (n+1) =\frac {n*(n+1)}{2} $$
Is what am I doing also proof and equivalent to the other solution given above, or am I completely lost?
Thanks!
Assuming $$\sum_{k=1}^nk=\frac{n(n+1)}{2}$$ you should check that $$\sum_{k=1}^{n+1}k=\frac{(n+1)(n+2)}{2},$$ but you start from this equation. You must not assume the assertion of the theorem.
The proof given in a picture you show us goes exactly in this direction.