Camel Transporting Bananas

How many bananas can be transported?

You may have seen examples of this problem on the internet. The usual way the problem is posed is the following:

You have 3000 bananas that you want to transport to a town 1000 kms away, using only a camel as the mode of transportation. The camel can carry a maximum of 1000 bananas at a time but eats one banana every km it travels while carrying bananas. If the camel is not carrying anything, he does not need to eat. What fraction of bananas can you transport?

The problem is not trivial because if you load up the camel with 1000 bananas each time, he will have eaten all of them by the time you get to the end of the journey.

We will solve this problem as it is usually posed and and then use it to find an exact solution to the general case where the camel can carry a load c, the distance to cover is d and the number of initial bananas is N = c x M. The questions we will ask are: How many bananas can be transported for finite N and what fraction of bananas can be transported in the limit when N tends to infinity?

The solution can be found here.