Title
Spreading Rumors
Problem Statement
A rumor starts by someone telling the rumor to two people on May 1. Each of
those two people are responsible for telling the rumor to two others on the next
day (May 2).At this time seven people know the rumor. On May 3, the four people
who heard the rumor on May 2 must each tell two more people. It is important for
8000 people to hear the rumor by May 14. Is this outcome likely to occur if the
pattern for spreading the rumor continues as started?
On what day will approximately one-half of the 8000 have heard the rumor? On
what day will 256 new people be told the rumor? If the rumor process continues
until May 20,how many new people will hear the rumor on that day?
If the rumor process continues for n days, how many new people will be told the
rumor on the nth day? What will be the total number of people who know the rumor
on that day?
Investigation/Exploration of the Problem
Let's map out what we know:
On May 1, 1 person told 2 people, now 3 people know.
On May 2, each of the 2 new people told 2 more people, now 7 people know.
On May 3, the 4 people who heard the rumor on May 2 must each tell 2 more people, now 15 people know
And so on...
Written mathematically, we can say
s_{n}=2(s_{n-1})+1 s_{n-1}= # of people told the day before
Using the formula, we can say that the number of people told on May 1 can be written as s_{n} = (2*1) + 1= 3,
on May 2 : s_{n} = (2*3) + 1= 7
on May 3 : s_{n} = (2*7) + 1= 15
and so on...
The problem asks the following questions that a spreadsheet can help us to answer.
Click here for the spreadsheet
It is important for 8000 people to hear the rumor by May 14. Is this outcome likely to occur if the pattern for spreading the rumor continues as started? YES! 32,767 people know the rumor on May 14
On what day will approximately one-half of the 8000 have heard the rumor? 4095 people know the rumor on May 11
On what day will 256 new people be told the rumor? 256 people will know the rumor on May 8th because 255 people know the rumor on May 7th and they are each responsible for telling 2 people. As indicated in the spreadsheet, 511 people know the rumor on May 8. If we subtract 511 from 255 (the number of people who know on May 7), we get 256. Thus, 256 NEW people will have head the rumor.
If the rumor process continues until May 20,how many new people will hear the rumor on that day? 2,097,151!!!!WOW!
If the rumor process continues for n days, how many new people will be told the
rumor on the nth day? What will be the total number of people who know the rumor
on that day?
s_{n}=2(s_{n-1})+1
s_{n}= # of people told on the nth day and s_{n-1}= # of people told the day before
.
Author &
Contact
Laura Thomas
Memorial Middle School
Conyers, GA