Half-Life of a drug

Question

The half-life of a certain drug is 5 days. That is, 50% of the drug has left the body after 5 days. Let's say someone takes 305 mg of this drug once every 5 days. How many mg of the drug remains in the body just after the eighth dose? Round your answer to the nearest hundredth.

So if they take 305 mg of this drug then after five days they have 152.5 mg of the drug left. Then they take the next dosage which adds on another 305 mg. Then I'm getting confused. Because the first dosage would be over after the five days and leave the body with 152.5 mg just like at the end of the first. I'm probably wrong and very confused but is even part of what I'm saying correct?

Answer 1

After $1$st dose: $305$ mg
After $2$nd dose: $305 + 305 \cdot 0.5=305 \cdot (1+0.5)$
After $3$rd dose: $305 + 305 \cdot (1+0.5) \cdot 0.5=305 \cdot (1+0.5+0.5^2)$
See the pattern?

Answer 2

After one dose: $305mg$

After two: $305 \times \frac 12 + 305 = 457.5mg$

After three: $457.5 \times \frac 12 + 305 = 533.75mg$

$\vdots$

After eight: $604.04mg$