The pond and the plant

A little water plant grows on a pond. Each day it doubles its cover of the water, and on day thirty it covers the whole pond.

A water plant doubles its cover of the pond each day and covers all of it on day $30$.

cover $C(d)$: $C(d+1)=2C(d),C(30)=1$.

The question: on which day is half the pond covered? Not day fifteen. Since the plant doubles every day, the day before it covers everything it covers exactly half. The answer is day twenty-nine.

When is half covered? On day $30-1=29$, since one more doubling gives everything: $\frac{1}{2}\times 2=1$.

$C(29)=\frac{1}{2},2C(29)=C(30)$.

Go back one more day and a quarter is covered, then an eighth. On day twenty, the plant covers only one part in one thousand and twenty-four of the pond, so little that you would not notice it.

Cover on day $d$ is $2^{d-30}$. Day $28$: $\frac{1}{4}$. Day $27$: $\frac{1}{8}$. Day $20$: $2^{-10}=\frac{1}{1024}$, hardly visible.

$C(d)=2^{d-30},C(20)=2^{-10}=\frac{1}{1024}$.

The lesson: with doubling, nothing seems to happen for a long time and then everything happens in the last days. If you wait until half the pond is covered, you have one day left.

Doubling: almost nothing for a long time, then everything in the last days. Waiting until half is covered leaves $1$ day.

$C(d)=\frac{1}{2}\Rightarrow 30-d=1$ day.