A striking example of an exceedingly fast build-up of some small quantity when repeatedly doubled is the famous legend about the award to be given to the discoverer of chess.* Here is [another] example, less famous. [I'm doing the first part only. Ed.]
The infusorian paramecium divides in half on the average every 27 hours. If all newly born infusorians remained alive, how long would it take for the progeny of one paramecium to fill up a volume equal to that of the Sun?
Starting data: the 40th generation of a paramecium, when none perish, occupies one cubic metre; we take the volume of the Sun as equal to 1027 metres.
* See my book Figures for Fun, Mir Publishers, Moscow.
(Source: Algebra Can Be Fun by Yakov Perelman)
The author solved it a bit different but here's what I did. I started with the equation:
\[ 2^x = 10^{27} \]
Where $x$ is the required number of doubling periods, starting from the initial cubic meter volume. I then used logarithms to obtain $x$:
\begin{align*}
\ln(2^x) &= \ln(10^{27}) \\
x \ln(2) &= 27 \ln(10) \\
0.69 x &\approx 62.1 \\
x &\approx 90
\end{align*}
So the required number of doubling periods is almost exactly 90. I added 40 to $x$ to take into consideration the initial growth to cubic meter size then multiplied by $\frac{27}{24}$ to get days and I got a result just a little short of the author's of 146.25 days.
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