1. ## Geometric series question

A large school board established a phone tree to contact all of its employees in case of emergencies. Each of the 3 superintendents calls 3 employees who each in turn calls 3 other employees and so on. How many rounds of phone calls are need to notify all 9840 employees?

A large school board established a phone tree to contact all of its employees in case of emergencies. Each of the 3 superintendents calls 3 employees who each in turn calls 3 other employees and so on. How many rounds of phone calls are need to notify all 9840 employees?

After $\displaystyle n$ rounds, the number of people contacted (including the original $\displaystyle 3$) is:
$\displaystyle 3 + 3\times3+3\times3^2+3\times3^3...+3\times3^{n-1}$
which is a Geometric Series with:
first term, $\displaystyle a = 3$; common ratio $\displaystyle r = 3$.
The sum to $\displaystyle n$ terms, $\displaystyle S_n$ is given by:
$\displaystyle S_n = \frac{a(r^n-1)}{r-1}$
$\displaystyle =\frac{3(3^n-1)}{2}$
We want $\displaystyle S_n \ge 9840$. So:
$\displaystyle \frac{3(3^n-1)}{2}\ge9840$

$\displaystyle \Rightarrow 3^n-1\ge6560$

$\displaystyle \Rightarrow n = 8$