Question

As part of a social experiment, Joey sent an email to 2 friends. The email asks the person reading it to forward it to 3 people. The table shows the distribution of the email, assuming that everyone who recelves the email forwards it to 3 new people.Which formula gives the number of people receiving the email anbased on the number of iterations, n?

Answer 1

Let's take a look at what happens for every iteration, so we can identify a common pattern that allows us to write down a formula for the n-th iteration.

Let P be the number of people receiving the email.

First iteration (n=1)

Joey sent an email to 2 friends

$P=2$

Second iteration (n=2)

Those two friends then send it to 3 friends each:

$\begin{gathered} P=2\cdot3 \\ P=2\cdot3^1 \end{gathered}$

Third iteration (n=3)

Those friends then send it to 3 friends each:

$\begin{gathered} P=2\cdot3\cdot3 \\ P=2\cdot3^2 \end{gathered}$

Fourth iteration (n=4)

Those friends then send it to 3 friends each:

$\begin{gathered} P=2\cdot3\cdot3\cdot3 \\ P=2\cdot3^3 \end{gathered}$

Now the pattern becomes clear. For every iteration, we have a 2 multiplying a 3 that has a power equal to the number of the iteration minus one. This way,

n-th iteration:

$P=2\cdot3^{n-1}$

In terms of the table,

$a_n=2\cdot3^{n-1}$