Question

the population of humorville is 8,500 people. in this town, jokes travel fast. in one hour, each person who hears a joke tells three other people who have not heard it, and tells no one else. last frieday, a visitor from out of town told the police chief a new joke at 10:00 a.m. how long did it take for everyone in humorville to hear the joke?

Answer 1

@Hero

Answer 2

@amistre64

Answer 3

so, what are your thoughts?

Answer 4

my thoughts is that this is an exponent problem @amistre64

Answer 5

it is an exponent yes

have you tried making a list of values?

h0 = 1
h1 = 1 + 3
h2 = 1 + 3 + 3(3)
h3 = 1 + 3 + 3(3) + 3(3)(3)

etc ....

Answer 6

no i havent @amistre64

Answer 7

well, making a list helps to see the process

Answer 8

what does my list suggest to you?

Answer 9

This is a geometric sequence which is exponential, but it's more than that.
In other words, you need to find after which period, the sum of the sequence reaches 10,000.

Answer 10

to add and multiply @amistre64

Answer 11

doesnt the number of people for a given hour look like a geometric sum? if so, how do we solve a geometric summation?

Answer 12

\[h_n=\sum_{k=0}^{n}3^n\]
when does this equate to the entire population?

Answer 13

3^k that is

Answer 14

wait im confused @amistre64

Answer 15

i hope you get more specific than that

Answer 16

i dont understand how to solve a geometric summation

Answer 17

well, with any luck you have a formula laying about for it

Answer 18

the other method is to simply do something nifty

Answer 19

S = 1 + r + r^2 + r^3 + r^4 + ... + r^n
-rS = - r - r^2 - r^3 - r^4 + ... + r^n - r^(n+1)

when we add these together, almost everything goes to 0

Answer 20

S = 1 + r + r^2 + r^3 + r^4 + ... + r^n
-rS = - r - r^2 - r^3 - r^4 + ... + r^n - r^(n+1)
---------------------------------------------
(1-r)S= 1 + 0 + 0 + 0 + 0 + ... + 0 -r^(n+1)

out goal is to find S so, divide both sides by 1-r

\[S=\frac{1-r^{n+1}}{1-r}\]

Answer 21

so now we have a formula to determine the summation of any nth hour.

how do we determine the solution to the problem now?