Chiu Chang Suann Jing/Examples/Example 10

Example of Problem from

 * Of $2$ water weeds, one grows $3$ feet and one grows $1$ foot on the first day.
 * The growth of the first becomes every day half of that of the preceding day
 * while the other grows twice as much as the previous day.
 * In how many days will the two grow to equal heights?

Solution

 * $2 \frac 6 {13}$ days, at which time they will grow to $4 \frac {11} {13}$ feet.

Proof
The growth of the $2$ plants is governed by a geometric sequence.

Let the height of the two plants after the first day be $a_1$ and $a_2$ respectively.

Let the common ratio of the growth rates of the two plants be $r_1$ and $r_2$ respectively.

Let $d$ be the number of days after which they reach the same height.

We have:

Thus we have that $2^d = 1$ or $2^d = 6$.

Hence either $d = 0$, which does not work, or $d = \log_2 6$.

This gives $d \approx 2.585$, and a common height of $5$ feet.