Title

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

sn=2(sn-1)+1        sn-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 sn = (2*1) + 1= 3,

on May 2  : sn = (2*3) + 1= 7

on May 3 :  sn = (2*7) + 1= 15

and so on...

• 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?

sn=2(sn-1)+1

sn= # of people told on the nth day    and    sn-1= # of people told the day before

.

Author & Contact
Laura Thomas

Memorial Middle School

Conyers, GA