The formula is given without explanation. You can rederive it without too much trouble.

You can't use the common formula for compound interest because users are assumed to send invites only once. Compound interest would assume they were sending invites continuously over the cycles.

If you look at the number of users added each cycle (the top row) you can see that it doubles each time. It's given by c0 times K^i, where c0 is the initial number of customers, K is the virality coefficient (2 in this case), and i is the i-th cycle.

Adding each of the terms up to i to the original c0 gives the total number of customers after i cycles. So you get a sum:

  sum over i from 0 to N of (c0 * K^i)

which using an exponential sum formula (http://mathworld.wolfram.com/ExponentialSumFormulas.html) gives:

  c0 * ((1-K^(N+1)) / (1 - K))

Multiply by -1/-1

  c0 * ((K^(N+1) - 1) / (K - 1))

and N is the number of cycles (given by t/ct in the slides)

  c0 * ((K^(t/ct+1) - 1) / (K - 1))