# How can solve this problem of geometric-progression?

A city has $29,524$ inhabitants. One listen about news. After an hour, this man announces to three of his neighbors. In one hour, each neighbors announces to three neighbors.

                                x <- Starter
-----------------------------------------
|                   |                   |
x                   x                   x
----------------       -----------        -------------
|       |      |       |    |    |        |     |     |
x       x      x       x    x    x        x     x     x
.. And so succesively.. (the mechanism of difussion.. only which here is based a reason)


These neighbors repeated communication under the same conditions. How long will it take to notify all the inhabitants of the city about the news?

I thought I'd convert $S_n$ to $S_n = 29,524/3$

$a_1 = 1$, $a_2 = 3$, $a_3 = 9$

$\frac{3^n-1}{3-1}=29524$ $\implies 3^n=59049$ Taking log to the base 3 on bothsides $n=\ln_3(59049)=9$ So the right answer is 9 hours
You're $a_i$ represent the number of new people being informed. The total number after $k$ iterations is $\sum_i^k 3^k$ which has a well known formula. Setting that formula equal to the number of people in the city, and then solving for $k$ produces your answer.
Hint: it should use the $\log_3(x)$ function