You are trying to prove that ?
Your fundamental problem is that this isn't true!
When n= 2, while .
"Plug in" n = 1. This then reads 1 = 1, good so far. Next try n = 2: 1 + 3 = 13 + 23, which is of course false. So your formula is not true for all n and so certainly can't be proved by induction.
The above is the correct formula and can be proved by induction. I suggest you try it.
It suddenly occurred to me that you probably copied the problem incorrectly. What is true:
This formula is amenable to induction. Try it.
For each line 1) - 4) (and letting the line number be ), I noticed that the first number in the sum is:
Now, we then are adding the next consecutive integers. As you observed, the sum is shown to be equal to . So, we may state this as:
Does this make sense?
If I was going to prove this by induction, this is what I would do. We have already shown the base case is true, so the induction hypothesis is the so-called conjecture:
Let's simplify a bit:
As the inductive step, I would add :
We have derived from thereby completing the proof by induction.