If the supplement stopped decaying one day after being ingested, then $25/64$ mcg would remain in his body for each day he takes the supplement, and your answer would be correct.
However, that's not quite what the problem implies (you are correct about the $12$th dose being taken on the $12$th day, though).
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The supplement continues to decay after the first day, so two full days after he takes the first dose, the first dose would have decayed by another factor of $64$, while the second dose would only have decayed by a single factor of $64$, meaning that he would have $$\frac{25}{64^2}+\frac{25}{64^1}$$ mcg left in his body immediately before he takes the third dose, and $$\frac{25}{64^2}+\frac{25}{64^1}+\frac{25}{64^0}$$ mcg of the supplement in his body immediately after taking the third dose. In other words, the amount of the supplement left in his body (in mcg) immediately after taking the $n$th dose can be expressed as $$25\sum_{i=0}^{n-1}\frac{1}{64^i}$$