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A rumor spreads through a city. At \(t=0\), one person knows the rumor. During the first hour, that person tells three people. Each person who newly hears the rumor takes exactly one hour to tell three other people who have not heard it.
a) How many people hear the rumor for the first time during hour \(5\)?
b) How many people have heard the rumor in all after \(8\) hours, including the first person?
c) After how many full hours have more than \(250{,}000\) people heard the rumor?
Hints
- Determine how many new people are added at each stage.
- For the total, add all groups from the initial person through the stated hour.
- Recognize the resulting geometric series.
- For part c, test consecutive whole-number powers of \(3\).
Solution
1. The number of people who hear the rumor during hour \(n\) is \(3^n\). During hour \(5\), \(3^5=243\) people hear it for the first time.
2. The total after \(n\) hours is a geometric sum: \(1+3+3^2+\cdots+3^n=\frac{3^{n+1}-1}{2}\).
3. After \(8\) hours, the total is \(\frac{3^9-1}{2}=9841\).
4. After \(10\) hours, the total is \(\frac{3^{11}-1}{2}=88{,}573\), which is below \(250{,}000\). After \(11\) hours, the total is \(\frac{3^{12}-1}{2}=265{,}720\), which is above \(250{,}000\).
Answer
a) \(243\) people
b) \(9841\) people
c) \(11\) hours
