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Thread: Geometric series question

  1. #1
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    Unhappy 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?

    The back of the book says the answer is 8 rounds but I have no idea how to get that Please help
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  2. #2
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    Hello needhelpplease1
    Quote Originally Posted by needhelpplease1 View Post
    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?

    The back of the book says the answer is 8 rounds but I have no idea how to get that Please help
    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:
    $\displaystyle \frac{3(3^n-1)}{2}\ge9840$

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

    $\displaystyle \Rightarrow n = 8$
    Grandad
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