Question

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?

Answer 1

`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`