Geometric series world problem from khanacademy of supplement doses

Question

Niklas takes a dose of 25 micrograms of a certain supplement each day. The supplement has a half life of 4 hours, meaning that $1/64$ of the supplement remains in the body after each day. How much of the supplement is in Niklas's body immediately after the $12th$ dose? Round your final answer to the nearest hundredth.

Answer: $25+25*(1/64)+25*(1/64)^2+...+25*(1/64)^{11}$

Why is that make sense? In my logic it should be

$((((25*(1/64)+25)*1/64)+25)*1/64 +...+25)*1/64$

And that make sense after first dose remaining ($25*1/64$), then he takes next dose and after remaining (($25*1/64$first dose remaining$+25$second dose)$*1/64$remaining after first dose remaining + second dose and so on) Does anyone can clarify me where I go wrong and why original answer make sense?

Answer 1

You should just try expanding out what you have. If you keep careful tabs on the number of doses, it'll come to exactly the same expression as what was stated in your answer key.

It's easy to see what's going on when you look at concentrations immediately after dose for a small number of doses:

$1$ dose: $25$

$2$ doses: $25\cdot \frac{1}{64} + 25$

$3$ doses: $(25\cdot \frac{1}{64} + 25)\cdot \frac{1}{64} + 25$

Already you can see the expression for the third dose resembles your expression on a smaller scale. When you expand it out you get:

$(25\cdot \frac{1}{64} + 25)\cdot \frac{1}{64} + 25 = 25\cdot (\frac{1}{64})^2 + 25 \cdot \frac{1}{64} + 25$

which is similar to the expression in your answer key (a geometric series).

Instead of a tedious expansion, an insight that allows you to immediately "see" how they got to a geometric series is to think of each dose being acted on independently and sequentially by metabolism.

So, the first dose would have had $11$ days "worth" of metabolism, which would have reduced its final concentration to $25 \cdot (\frac{1}{64})^{11}$.

The second dose would have had $10$ days "worth" of metabolism, which would have reduced its final concentration to $25 \cdot (\frac{1}{64})^{10}$.

And so on, until you get to the $12$-th dose, which remains "brand new" at exactly $25$.

Now just sum those up!