Frog and the Well

5-3=2 100/2 = 50 50-1=49

In one day and one night, he climbs up a total of $2$ meters. In $48$ days and $48$ nights, he climbs up a total of $2\times 48=96~meters$ . Therefore, he needs another day to get out the well. So the number of days is $49$ and the number of nights is $48$ .

It takes 2 leaps in a day, 5 up and 3 down. It moves a total of 5-3 = 2m a day.

Let total no of days it took be "n". $\implies 2*(n) = 100 \\ \implies n = 50\text{ max-days.}\\ \text{on 48}^{th} \text{ day, it will at a height of 48 } \times \text{ 2 = 96m. So, in the next day's dawn, it'll climb up 5m which gives, 96 + 5 = 101.} \\ \text{Therefore, in the 49th day, it'll totally climb up the river.}$

So, how did I find that 48th day?

We know that it must reach 100m, but we have sense that it cannot come down to well after it climbed up the well. So, max-distance it traveled in before leap to climb up well is the difference between 100 and 5, ie, 100 - 5 = 95. So we have the distance, now using the formula $\implies 2n = 95 \\ \implies \lfloor n \rfloor = 95/2 \\ \implies n = \lfloor 47.5 \rfloor\\ \implies n = 48 days$

Therefore the answer is $\fbox{49 days}$