I am studying finance and I come across some formulas that I've been trying to get some intuition on. Here we have an equality of (simplified and applied to 3 periods without the Cash flow part) PV expressions:
$$\dfrac{1}{1+x}+\dfrac{1}{\left( 1+x\right) ^{2}}+\dfrac{1}{\left( 1+x\right) ^{3}}=\dfrac{1}{x}\left( 1-\dfrac{1}{\left( 1+x\right) ^{3}}\right)$$
I know that they are equal, but after several hours trying to go from the left expression to the right one I have not managed to do it.
Can someone walk me through (or point to) the steps needed to arrive at the right expression?
If you let $r=\frac1{1+x}$, the LHS is $r+r^2+r^3$.
Notice that $1-r=1-\frac1{1+x}=\frac{x}{1+x}=rx$.
Hence the right hand side should be $$r\cdot \frac{1-r^3}{1-r}=r\cdot \frac{1-r^3}{rx}=\frac{1-r^3}{x}$$