<s>[INST] Simplify expression involving fractions for mathematical induction proof. I have this expression
$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$
that I need to manipulate into looking like
$1-\frac{1}{k+2}$
as the final step in my mathematical induction proof process.
My idea was to go something like this:
\begin{align*}
1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)} &= 1-\frac{k+2}{(k+1)(k+2)}+\frac{1}{(k+1)(k+2)}\\
                                   &= 1-\frac{(k+2)+1}{(k+1)(k+2)}\\
                                   &= 1-\frac{(k+2)+1}{(k+1)(k+2)} \cdot \frac{k+1}{k+2}\\
                                   &= 1-\frac{1}{k+2}
\end{align*}
Is this algebraically correct?
 [/INST] The first part is right, but:
$$1-\frac{k+2}{(k+1)(k+2)}+\frac{1}{(k+1)(k+2)}$$
$$= 1 - \left(\frac{k+2}{(k+1)(k+2)} -\frac{1}{(k+1)(k+2)} \right)$$
$$= 1 - \frac{k+2 \color{red}{-} 1}{(k+1)(k+2)}$$
and you can continue from here.
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