I'm a bit of a maths noob so please bear with me with what is probably a really dumb question, but I could really do with some help - I'm self-learning at home.
I'm stuck on the question below from Discrete Mathematics for Computing (Peter Grossman) on proving a recursive definition by induction. I understand everything up to the highlighted line in yellow.
It asks to create a non-recursive version of the formula
$t(n) = t(n-1) + 2n - 1$
Writing down the first few terms indicates a squares sequence, so a non-recursive form is:
$t(n) = n^2$
The question asks to then prove this is correct. The base is shown in the question, t(1) = 1 so the base is:
$t(1) = 1^2$
I then assume (inductive hypothesis) that
$t(k) = k^2$
I then need to prove that it's true for every successive term, i.e. with n = k+1. So I thought the next step would be:
$t(k+1) = (k+1)^2$
But instead the book jumps back to the original, non-recursive formula at this point (yellow line), and I just don't get that bit - surely we are now proving t(n) - t(n-1) + 2n -1 instead of t(n) = n^2? I can't assume these are equivalent at this point, as we are only conjecturing that they are the same... confused!