Question

Five girls decide to spread a rumor. Within the first hour, each girl tells four other
people. Then within each subsequent hour, each person that knows the rumor
continues to tell four more people about it. Let x = the number of hours, and y = the
number of people who know of the rumor. After five hours, how many people know
about the rumor? Set up the equation represented by this situation using the
variables above, and solve the problem, showing the calculation that you did to get
your answer.

Answer 1

Answer:

1705

Step-by-step explanation:

This is a sequence where the first term, a=5

Since each girl tells 4 other people, the number of people told after the first hour =20

Each subsequent hour, each person that knows the rumor  continues to tell four more people about it.

Therefore we have the sequence: 5, 20, 80,...

This is a geometric sequence in which the:

• First term, a=5
• Common ratio, r=4

To determine the number of people, y who have heard the rumor after x hours, we simply find the sum of the geometric sequence.

$\text{Sum of a Geometric series},y=\dfrac{a(r^x-1)}{r-1} \\ After 5 hours, x=5\\y=\dfrac{5(4^5-1)}{4-1} \\y=\dfrac{5(1023)}{3}\\\\y=1705$

Therefore, the total number of people who know about the rumor after five hours is 1705.