# Thread: help with equation in alternate form

1. ## help with equation in alternate form

I'm told that the following two equations are equal, and asked to write the steps to manipulate the first equation into the second.

$1 - \frac{1}{k + 1} + \frac{1}{(k+1)(k+2)}$

to

$1 - \frac{1}{k + 1}\bigg[1 - \frac{1}{(k+2)}\bigg]$

I've tried many different ways of writing the first equation, but I can't seem to get it to work. Could someone point me in the right direction?

2. please don't edit your problem if you type something wrong then type it in another post.
\\edit
just open brackets

3. Perhaps it would be easier to show you how to get from the bottom to the top instead of the other way around.
$1 - \frac{1}{(k+1)} + \frac{1}{(k+1)(k+2)} = 1-\frac{1}{(k+1)} \bigg[1-\frac{1}{(k+2)}\bigg]$

$1 - \frac{1}{(k+1)} + \frac{1}{(k+1)(k+2)} = 1 + \bigg[\bigg(-\frac{1}{(k+1)}*1\bigg) - \bigg(-\frac{1}{(k+1)}*\frac{1}{(k+2)}\bigg)\bigg]$

$1 - \frac{1}{(k+1)} + \frac{1}{(k+1)(k+2)} = 1 + \bigg(-\frac{1}{(k+1)}\bigg) + \bigg(\frac{1}{(k+1)(k+2)}\bigg)$

Does that make it easier? I think the most common error people would make there is not distributing (or removing) the negative along with $\frac{1}{(k+1)}$.

4. Thanks so much! That makes sense.