Calculating time taken for a set percentage of a reaction to occur based on half-life
Ryan showed that the rate constant is given by $k=\frac{ln(2)}{t_{1/2}}$.  So the concentration at time t is given by
$$C=C_0\exp(-kt)=C_0e^{-ln(2)\frac{t}{t_{1/2}}}$$But,
$e^{-ln(2)}=\frac{1}{2}$.  Therefore, $$C=C_0\left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}$$Substituting $n=\frac{t}{t_{1/2}}$ gives
$$C=C_0\left(\frac{1}{2}\right)^n$$