Gems for profit

The blue gem makes €30 income for every €20 spent. That means, €1.5 is earned for every euro spent for blue gems.

This ratio for the pink gem is 5 at the start, as the initial sale price is €5000 and the buying price is €1000. As the price is lowered by 20% for each next gem, the n:th pink gem's price in euros is $5000\times{0.8^{n-1}}$ . The income per euro spent ratio is $\frac{5000 \times {0.8^{n-1}}}{1000}=5\times{0.8^{n-1}}$ .

The most profit will be earned when the seller buys pink gems until the next gem he/she would buy, would give less income than €1.5 for each euro spent (the stable income per euro spent ratio for blue gem). We get the following equation:

$5\times{0.8^{n-1}}=1.5$

$0.8^{n-1}=0.3$

$n-1=\log_{0.8}{0.3}$

$n=\log_{0.8}{0.3}+1$

$n=6.39511...$

This means, most profit will be made with 6 pink gems bought with €6000, and the other €4000 spent for blue gems:

The income earned with pink gems:

$\sum_{n=1}^6(5000\times(0.8)^{n-1})=\frac{5000(1-0.8^6)}{0.2}=18446.4$

The income earned with blue gems:

As blue gems make €1.5 income for every euro spent, the income in euros is $4000\times1.5=6000$

The total profit is income minus expenses: $18446.4+6000-10000=\boxed{14446.4}$ .