Question

A farmer planted a lotus plant in the village pond. One fine day, to the happiness of the villagers, one lotus flower had bloomed.
The next day the saw two flowers. The third day four flowers were seen. Everyday they saw double the number of flowers then the
previous day. On the $30th$ day the villagers found that half of the pond was filled with lotus flowers. How many days more will
it take to fill the poind with lotus flowers? In how many days the pond was quarter full with lotus flowers?

Answer 1

Let total flowers be a to fill pond.

$\frac{a}{2}=1+2+4+......+2^{30-1}\frac{a}{2}=2^{30-1}a=2^{31}-2$

For $31$ days flowers cannot $=2^{31}-1$

Just one more day to fill pond completely

Number of flowers to fill quarter pond $==2^{29}1+2+4+......+2^{n-1}=2^{29}{2}^n-1=2^{29}=>n=29(2^{29}>2^{29}-1)$

It will take $29$ days to fill quarter pond.