MHB Equilibrium Problem: Landscaping Business|Accounts & Maintaining 500

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To determine the equilibrium number of accounts for a landscaping business, the formula nx = 500 is essential, where n is the number of new accounts added weekly and x is the average duration of account retention in weeks. As accounts are added, some will drop off, creating a dynamic balance that must be maintained to reach the target of 500 accounts. The discussion highlights that if five new accounts are added weekly, the actual number of accounts will be less than 250 after 50 weeks due to customer turnover. The relationship between the number of accounts added and their retention is inversely proportional, complicating the calculation of equilibrium. Understanding this balance is crucial for effective business capacity planning.
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I own a landscaping business with accounts paying regularly. If I add 5 new accounts a week and on average an account stays a customer for one year before quitting, what would my equilibrium number of accounts be? How long to reach equilibrium? how many do I need to add per week to maintain at a certain number, say 500?

This is a real situation for me trying to plan business capacity in the future. A formula for the general case would be much appreciated, n accounts added, stay on for x number of days before quitting, etc.
 
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Suppose you start at day zero, no accounts, and then as you progress, you have an average of $n$ new accounts per week who each stay with you for $x$ weeks. At the end of the first $x$ weeks, you then have $nx$ accounts, and thereafter, you have the same number of accounts dropping as you are adding, so your equilibrium would be $nx$. In order for this to be 500, we require:

$$nx=500$$

There are an infinite number of solutions...as $n$ increases, then $x$ decreases (and vice versa), as there exists an inverse relationship between the two.
 
At the end of the first x weeks I don't think I have nx accounts. After say 50 weeks, given n = 5 it would not be 250 accounts, it must be less than 250 accounts.

If each account stays a year on average some number would have quit before 50 weeks so it must be less than 250.

My math is rusty but it's more complicated, I don't know if it's a diff eq problem, been awhile for me. Thx
 
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