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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  
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    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 $S_n = 29,524/3$

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

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3 Answers 3

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$\frac{3^n-1}{3-1}=29524$ $\implies 3^n=59049$ Taking log to the base 3 on bothsides $n=\ln_3(59049)=9$ So the right answer is 9 hours

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First, you need to specify if a person who's already heard the news can be told it again. If everyone finds 3 new "neighbors" to tell it to after they heard it (or until everyone has heard it), then you are simply looking a solution to a logarithmic problem, since logarithms solve exponential (and geometric) problems similar to what you mentioned. Try to use the geometric sum formulas.

However if neighbors are allowed to hear things multiple times, then it will take longer than that, since after a while, the same neighbors will hear things over and over. In this case, it could take any number of days (past the minimum from the previous paragraph).

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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

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