Simon from Elizabeth College, Guensey and Andrei from Tudor Vianu National College, Romania both solved this problem using geometric series. Here is Simon's solution:

The equation to show stock levels in week n is:
Un = X + æ
ç
è 		1 - p
100
 ö
÷
ø 		Un-1.
Substituting for Un-1 and simplifying:
Un
=
X + æ
ç
è 		1 - p
100
 ö
÷
ø 		æ
ç
è 		X + æ
ç
è 		1 - p
100
 ö
÷
ø 		Un-2
=
X + æ
ç
è 		1 - p
100
 ö
÷
ø 		X + æ
ç
è 		1 - p
100
 ö
÷
ø 		2

 
 Un-2.
Originally there was no stock so the initial condition is U0 = 0. To make the expressions easier to read we write
s = æ
ç
è 		1 - p
100
 ö
÷
ø

.
Writing Un in terms of U0:
Un = X(1 + s + s2 + ... + sn-1)+ snU0 = n-1
å
r=0 
 Xsr.
Summing this geometric series
Un = X(1-sn)
1 - s
 = 100X
p
 æ
ç
è 		1 - æ
ç
è 		1 - p
100
ö
÷
ø 		n

 
 ö
÷
ø 		.
To find the limit of Un over a very long period of time, as (1 - p/100) < 1, we have

lim
n® ¥ 
 æ
ç
è 		1- p
100
 ö
÷
ø 		n

 
 ® 0
and so

lim
n® ¥ 
 Un = 100X
p
 .
The company should ensure that they have enough warehouse space for
100X
p

items. However, the chances are that sales and production will vary during the year, and therefore they should ensure that they have a buffer zone so that they can have more stock if necessary. Also, they are unlikely to be in business for an infinite length of time and therefore should cater said limit to their needs based on the financial situation.