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Simple algebra really:
1+r+...r**n=r(1/(r+1)+r+...r**(n-1)) - Factor out r as Cadd says.
Divide by (1-r)/(1-r) ---This is effectively 1 so doesn't change anything.
[r(1/(r+1)+r+...r**(n-1))] x (1-r)/(1-r)=
{[r(1/(r+1)+r+...r**(n-1))]-[r+r**2.....+r**(n+1)]}/(1-r)
Above you have just "opened up the brackets and multiplied by 1 and by -r.
={(1+r+r**2....+r**n)-(r+r**2+ .....r**(n+1)}/(1-r)
Cancell all + and - terms above:
{1-r**(n+1)}/(1-r)
QED
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