How many eggs did he have when he began his deliveries?

Question

 A farmer has to make eight stops and delivering eggs. He begins with exactly the number of eggs he needs for these eight deliveries. At the first stop he delivers half of the eggs he has +1/2 of an egg. At each of the next seven stops he delivers half of the remaining eggs +1/2 of an egg. When he's finished he has no eggs left, and none have been broken when making the deliveries.

 

I'm looking not for the answer but for the formula to find what the answer is. I need to figure out how to write the sequencing equation

Answer 1

Let a(i) = the number of eggs the farmer has after the "i"th stop.  

Note that a(i-1) is the number of eggs the farmer has before the "i"th stop.  

So a(0) = the number of eggs the farmer begins with (before the first stop).  

 

At the "i"th stop the farmer starts with a(i-1) eggs, delivers half of these +1/2 of an egg, and ends up with a(i) eggs, so the relationship between a(i-1) and a(i) is given by   

 

a(i-1) - [ (1/2)*a(i-1) +(1/2) ] = a(i)  

 

The expression within the [ square brackets ] represents the delivery of half the remaining eggs +1/2 of an egg.  Solving this equation for a(i-1) gives  

 

(1/2)*a(i-1) - (1/2) = a(i)

a(i-1) - 1 = 2*a(i)

a(i-1) = 2*a(i) + 1 

 

We are given that a(8) = 0 since he has no eggs left after the eighth stop.  You can use this to recursively figure out each of the previous a(i) until you get to a(0), which is the number of eggs he started with.