How does step 1 convert to step 2? We don't know what n is so I think step 2 is just making assumptions

The Blade Dancer · Nov 28, 2020, 9:53 PM

Module 1 Week 1 Day 1 Your Turn Part 1 Mini-Question

debbie · Nov 28, 2020, 9:51 PM

@the-blade-dancer The second step looks quite different from the first step! It's because for the first two fractions, we have multiplied the top and bottom by a factor.

For the first fraction, we have multiplied the top and bottom by $n + 1.$

$\frac{1}{\textcolor{red}{n}} = \frac{n + 1}{\textcolor{red}{n} (n + 1) }$

For the second fraction, we have multiplied the top and bottom by $n.$

$\frac{1}{\textcolor{red}{n+1}} = \frac{n}{\textcolor{red}{(n +1)} n }$

For the third fraction, we've done something different: we have factored the bottom. (Factor just means to write as a product of two terms, like $12 = 3 \times 4.$ )

$- \frac{1}{n^2 + n} = - \frac{1}{n(n+1)}$

Why are we trying to do this? To make the bottoms of the fractions all equal to $n(n+1).$ Then we can add the fractions together very easily, and luckily we have some terms cancel out, so we get $0$ on the top, which is a little fortunate!

$\frac{n + 1 - n - 1}{n(n+1)}$

$= \frac{0}{n(n+1)} = \boxed{0}.$