1. ## Tricky Story Question

Pretend you start a chain letter. You send it to seven friends, who each send it to seven friends, and so on. If each person sends it to seven new people who have never seen it before, how many people have seen it by the eighth generation (you count as the 1st generation).

2. Originally Posted by Mr_Green
Pretend you start a chain letter. You send it to seven friends, who each send it to seven friends, and so on. If each person sends it to seven new people who have never seen it before, how many people have seen it by the eighth generation (you count as the 1st generation).

number in the 0-th generation (that's you) n_0=1
number in the 1-st generation n_1=n_0*7=7
number in the 2-nd generation n_2=n_1*7=49

and so on.

So we may evaluate this chain multiplying by 7 for each generation, or we
can observe that this gives us 7^r in the r-th generation.

So the number who have seen it by the r-th generation is the sum of the
number in each of the generations up to and including the r-th:

N_r=n_0 + n_1 + .. + n_r = 1+ 7 + 7^2 + ... + 7^r.

This is a finite geometric series (or progression), who's sum is:

N_r= 1+ 7 + 7^2 + ... + 7^r = (1-7^9)/(1-7) = 6725601.

(you are expected to know that:

$S_k= 1 + \rho + \rho^2 + .. + \rho^k = \frac{1-\rho^{k+1}}{1-\rho}$

RonL