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using http://openstudy.com/study#/updates/504b9d7ce4b0985a7a58b494 I understand everything until

I don't get how the last form is "precisely what we were supposed to get" and how it proves it true

for n= 1, we have:
left side = 1^2 = 1
right side= 1*(6-3-1)/2= 1
so our proposition is true for n=1

ok so we have the basis proven true.. next we need to have n = k

in general, the mathematical induction principle requests another checking, for example n=2

Ok I understand why and how to do that

so, for n=2, we have:
left side = 1+ 4^2=1+16=17
right side = 2*(24-6-1)/2= 17

but what about n = k + 1?
how is the simplified form of n = k+1 proving it true???

first we need n = k and then we go to n = k+1 . That guy went forward to n+1

Ok

the whole string before (3k+1)^2 ends up being replaced by a proposition.

I think I'm more confused now

I also need to eat :/
I can't think without carbs.

thank you