Showing that $\frac{1}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$

Question

I have a simple development of an expression that I can not understand how it was made. The expression is:

$$\begin{align}\frac{1}{k(k+1)} & = \frac{1}{k} - \frac{1}{k+1} \end{align}$$

Can please someones show me how the left side is equal to the right side?

Thank you!

Answer 1

When adding two fractions, it is useful to use common denominator.

In your case, $k(k+1)$ is a multiple of both $k$ and $k+1$, so you can write:

$$\frac1k-\frac1{k+1}=\frac{k+1}{k(k+1)}-\frac{k}{k(k+1)}=\frac{(k+1)-k}{k(k+1)}=\frac{1}{k(k+1)}.$$

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A slightly different approach: If you are given $\frac1{k(k+1)}$ and you want to simplify it, you may notice that $\frac{k+1}{k(k+1)}=\frac1k$ and $\frac{k}{k(k+1)}=\frac1{k+1}$ are simpler expressions. So you can ask whether you can somehow write the numerator using $k+1$ and $k$. And you can: $1=(k+1)-k$.

So you get $$\frac1{k(k+1)} = \frac{(k+1)-k}{k(k+1)}=\frac{k+1}{k(k+1)}-\frac{k}{k(k+1)}=\frac1k-\frac1{k+1}.$$

Answer 2

Recall how fractions are subtracted: $$ \frac{a}{b} - \frac{c}{d} = \frac{a \cdot d}{b \cdot d} - \frac{b \cdot c}{b \cdot d} = \frac{a \cdot d- b \cdot c}{b \cdot d} $$ In the case at hand $a=c=1$, $b=k+1$ and $d=k$.