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

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

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 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!

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

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