Nails

A rider decided to buy a good riding horse, whose price was 10 000 €. The seller told him: "I will give you the horse for free. Pay me only for the nails with which the horseshoes are attached. For the first nail in the horseshoe pay me 1 cent, for the second 2 cents, for the third 4 cents, etc."
Was it a good deal if each horseshoe is attached with five nails? What was the price of the last nail?

Final Answer:

x =  0
a₂₀ =  5242.88 eur

Step-by-step explanation:

$$\begin{gathered} a=10000\text{eur} \\ GP \\ q=2 \\ {a}_1=0.01\text{eur} \\ n=5\cdot 4=20 \\ {s}_{20}=\Sigma a_{20}+a_{19}+a_{18}+\dots .+a_1+a_0 \\ {s}_{20}=a_1\cdot \frac{q^n-1}{q-1}=0.01\cdot \frac{2^{20}-1}{2-1}=\frac{41943}{4}=10485\frac{3}{4}=10485.75\text{eur} \\ {s}_{20}>a \\ x=0 \end{gathered}$$

$a_{20}=a_1\cdot q^{n-1}=0.01\cdot 2^{20-1}=5242.88\text{eur}$

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