# Recursive sequence and limits help

• March 4th 2010, 12:53 PM
Innit
Recursive sequence and limits help
Hi all,

I know this looks long, but please take the time to read it... You might actually find this problem quite interesting. I'm doing a fairly open ended maths investigation and have reached a bit of a road block.

Ok, so I begin with 100 thingies. The number of thingies increase by 90% every 6 minutes. However, over the period of one minute, 10 thingies are eliminated.

So I began to show this with calculations. I call the number of thingies A, and want to find a function for A in terms of time t.

So I look at the first few 6 minute intervals:
$A(6) = 100 \times 1.9 - 10 \times 6 = 130$
$A(12) = 130 \times 1.9 - 10 \times 6 = 187$
$A(18) = 187 \times 1.9 - 10 \times 6 = 295.3$

So basically I get a recursive sequence, where U_n is determined by U_(n-1).

However, you might notice that the above sequence is only an approximation, and reducing the time interval between each term increases the accuracy of my results. For example:
$A(1) = 100 \times 1.9^{1/6} - 1 \times 6 = 101.29$

Then repeating this up to A(6) will give a different result from above. So, the objective is to get as accurate as possible, and therefore reduce the time interval between each of my readings. If we call the time interval n, then I get the following function:
$A(t) = A(t-n) \times 1.9^{n/6} - 10n$

I find that if I reduce the value of n and graph the data points that I get, the data points slowly tend towards a specific exponential curve. However, I don't know how to figure out the equation of this curve. Do I use limits? How do I do this with a recursive sequence/function? I'm trying to find the continuous function that the data points tend towards.

Apparently there is a way of doing this, but perhaps I'm not aware of the method used in such situations (I've never really looked at recursive sequences in detail before). Any help would be appreciated!
• March 4th 2010, 09:58 PM
Innit
Any ideas?
Are there any methods for dealing with this kind of situation?
• March 6th 2010, 02:39 AM
Innit
Ok one last bump.

I've been told that I can try to solve this as a first order recurrence relation. Any ideas how to approach it that way?