Geometric series to calculate price

Question

I decided to add my extension to this question as a new question here. I am trying to represent the following as a geometric series equation:

1 item = $10
2 items = $((10*0.8)*0.9)
3 items = $((10*0.8)*0.9^2)
4 items = $((10*0.8)*0.9^3)

Answer 1

The first item costs $2+8*0.9^0$, the second and subsequent $8*0.9^{(n-1)}$. So the sum is $2+8(\frac {1-0.9^n}{1-0.9})$

Answer 2

$$ s = 10+( a r + a r^2 + a r^3 + \dots ) = 10 + ar( 1+ r+r^2+\dots )=10+ar.\frac{1}{1-r}, $$

where $a=10\times 0.8 =8$ and $r=0.9 $.

Added: In general, if

$$ s_n=a+ar+\dots+ar^n $$

then we can find a closed form for it. Multiply the above by $r$, we get

$$ r s_n = ar + ar^2+\dots+ar^{n+1}. $$

Subtracting the two equations gives rearranging

$$ s_n = \frac{a(1-r^{n+1})}{1-r}. $$