A rumour was spread by 4 man at 9:00am. In a period of 15 minutes,each of the 4 man told the rumour to another 3. For the following 15 minutes,each of the man told the rumour to another 3 who have yet to hear it. At 11:00am, how many man had heard it?

1 Answer
Nov 1, 2017

color(red)(13,120)

Explanation:

Note that there are 8 time periods (of 15 minutes each) between 9:00am and 11:00am

At time 0
the number of men with the rumour would be
color(white)("XXX")a_0=4

At time 1
the number of men with the rumour would be
color(white)("XXX")a_1color(blue)=4+3 *color(blue) 4

At time 2
the number of men with the rumor would be
color(white)("XXX")a_2=color(blue)(4+3 * 4) + 3 * (color(blue)(4+3 * 4))

In general at time n
the number of men with the rumour would be
color(white)("XXX")Sigma_(i=0)^n 4 * 3^n

This is the sum of a standard geometric series with a_0=4 and r=3

We know that
color(white)("XXX")Sigma_(i=0)^n a_0r^n=a_0((1-r^i)/(1-r))

Using the above values for a_0 and r
and with n=8
we have
color(white)("XXX")4 * ((1-3^8))/((1-3))

color(white)("XXX")=4 * (1-6561)/(-2)

color(white)("XXX")=2 * 6560

color(white)("XXX")=13120