Question

Assume you have n users and know that n grows by a fixed number of users c per day and viral growth rate of k per day, where k is expressed as a percentage of n.

How can you tell how many days it will take for the userbase to grow to size x, where x > n?

This is a compound interest problem, but I don’t know how to do it with the addition of the constant factor c.

Answer 1

Assuming you don’t mind fractional users… If k is expressed as a decimal (so a growth rate of 5% is k=1.05), then the formula is:

days=log_k(((1-k) x – c) / ((1-k) n – c))

For instance, suppose your initial userbase is 5; you grow constantly by 3 users per day, and virally by 5% per day; and your target is 35 users. Then

days=log_1.05(((-0.05)*35 – 3) / ((-0.05)*5 – 3)) = 7.78.

Running the process in Excel, you can see that, indeed, day 7 gives you 31.5 users, and day 8 36 users.

Derivation:

Denote the number of users after d days as n_d. Then:

n₁ = kn + c

n₂ = kn₁ + c = k( kn + c) + c = k²n + (k + 1)c

n₃ = kn₂ + c = k( k²n + (k + 1)c) + c = k³n + (k² + k + 1)c

…

n_d = k^dn + SUM_i=0,d-1(kⁱc)

Now, the SUM is a geometric series. The sum of that geometric series is easily derivable (or found on Wikipedia!) to be c(1 – k^d)/(1 – k).

So:

n_d = k^dn + c(1 – k^d)/(1 – k)

= k^dn + c/(1 – k) – ck^d/(1 – k)

= k^d( n – c/(1 – k)) + c/(1 – k)

So

k^d = (n_d – c/(1 – k)) / (n – c/(1 – k))

= ((1 – k) n_d – c) / ((1 – k) n – c)

So

d = log_k(((1 – k) n_d – c) / ((1 – k) n – c))