Compnay growth model

Hello everyone.
My task is to build a growth model for a company. The company offers services to businesses - and thus allowing the businesses to recruit new clients.
Assumptions:
- I started working with 10 businesses
- A business I start working with initially has 20 clients
- I'm recruiting 3 new businesses every month
- A businesses which is using my services will get 5 new client after every month of service (if he started getting my service on July 1st then on August 1st he will have 5 new clients)

I need to build a formula to calculate how much clients and how much businesses there will be in 6 months.

Here is how I started calculating manually:

MONTH 1:
10 (type A) businesses
20 clients (per business)
TOTAL - 10 businesses
TOTAL - 200 clients

MONTH 2:
10 (type A)businesses
25 clients (per business)
3 (type B) businesses
20 clients (per business)
TOTAL - 13 businesses
TOTAL - 310 clients

MONTH 3:
10 (type A)businesses
30 clients (per business)
3 (type B)businesses
25 clients (per business)
3 (type C) businesses
20 clients (per business)
TOTAL - 16 businesses
TOTAL - 435 clients

etc...


I need these to be variables that can be changed:
- number of businesses I started with
- how much new businesses I find every month
- how much new clients a businesses recruits after working a month with me
- how much clients a businesses initially has

pitkas said:

- number of businesses I started with : B[sub:214uezxx]0[/sub:214uezxx]
- how much new businesses I find every month b
- how much new clients a businesses recruits after working a month with me c
- how much clients a businesses initially has d

Click to expand...

1) C[sub:214uezxx]0[/sub:214uezxx] = dB[sub:214uezxx]0[/sub:214uezxx]
2) B[sub:214uezxx]n+1[/sub:214uezxx] = B[sub:214uezxx]n[/sub:214uezxx] + b
3) C[sub:214uezxx]n+1[/sub:214uezxx] = C[sub:214uezxx]n[/sub:214uezxx] + cB[sub:214uezxx]n[/sub:214uezxx] + bd

From (2), I get

4) B[sub:214uezxx]n[/sub:214uezxx] = B[sub:214uezxx]0[/sub:214uezxx] + nb

This changes (3) into

5) C[sub:214uezxx]n+1[/sub:214uezxx] = C[sub:214uezxx]n[/sub:214uezxx] + cbn + (cB[sub:214uezxx]0[/sub:214uezxx] + bd)

Then, (5) and (1) give

6) C[sub:214uezxx]n[/sub:214uezxx] = (cb/2) n[sup:214uezxx]2[/sup:214uezxx] + (cB[sub:214uezxx]0[/sub:214uezxx] + bd - cb/2) n + dB[sub:214uezxx]0[/sub:214uezxx]

In particular, after 6 months,

B[sub:214uezxx]6[/sub:214uezxx] = B[sub:214uezxx]0[/sub:214uezxx] + 6b, and
C[sub:214uezxx]6[/sub:214uezxx] = 15cb + 6cB[sub:214uezxx]0[/sub:214uezxx] + 6bd + dB[sub:214uezxx]0[/sub:214uezxx]

In your example, you gave :

* B[sub:214uezxx]0[/sub:214uezxx] = 10
* b = 3
* c = 5
* d = 20

So,

B[sub:214uezxx]n[/sub:214uezxx] = 10 + 3n
C[sub:214uezxx]n[/sub:214uezxx] = C[sub:214uezxx]n[/sub:214uezxx] = (15/2) n[sup:214uezxx]2[/sup:214uezxx] + (205/2) n + 200

In particular, after 6 months,

B[sub:214uezxx]6[/sub:214uezxx] = 28
C[sub:214uezxx]6[/sub:214uezxx] = 1085

Dear Dr.Mike - you have been very helpful - thank you very much!

I have made some adjustments to the formulas you gave and it works great, this is what I came to:

n Number of months to include in the report
A0 How many businesses I start with on the 1st month
B0 How many clients an initial business has
b How many new clients a business recruits every month
a How many businesses we recruit every month

Total businesses:
An=A0+(n-1)*a

Total clients:
Bn=(a*b/2)*(n-1)^2+(b*A0+a*B0-b*a/2)*(n-1)+A0*B0


There is another thing I would like to ask your assistance with, this is really complicated
I need to also calculate how much money the company earns in the end of each month

There is a difficult pricing structure that I need to follow.
Every single business that works with the company is billed monthly according to the amount of clients it has in total.
1. 20-49 clients = $36*client
2. 50-99 clients = $32*client
3. 100+ clients = $28*client

For example if on the 3rd month the company has:
20 businesses with 20 clients = 20*20*$36 = $14,400
10 businesses with 60 clients = 10*60*$32 = $19,200
2 businesses with 120 clients = 2*120*$28 = $6,720
* Then the company's revenue for the 3rd month is: $40,320

I have no idea how to implement this into the formulas so I'm hoping you can help


Thanks you,
Pete