I am currently stuck on an induction proof where I don't seem to understand why the positive 2(k + 1) becomes negative after giving both parts of the equation equal denominators. What is the reason for this? thank's a lot in advance!
If it is unclear what part of the equation I'm talking about, it is the 4th and 5th line.
For convenience let us adapt some shorter forms. Let $a = 2k+1$, $b=k+2$, c = $k+1$ and $x = k(k+1)(k+2)$. Now observe the forth equation $$\frac{3}{2}-\frac{ab}{x} + \frac{c}{x}.$$ Now let us take $-1$ out as common factor, then $$\frac{3}{2}-1\left(\frac{ab}{x} - \frac{c}{x}\right).$$ We then have $$\frac{3}{2}-1\left(\frac{ab -c}{x}\right).$$
I hope this helps.