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. Number of iterations Number of People Recelving the Email.

I can not see the table, so i will answer this in a general way.

Ok, let's see a rough image of what is happening.

First, Joey sends the email to 2 friends.

So first, 2 people have the email

Then each friend sends the email to 3 new people.

So now the number of people that has the email is:

2 + 2*3 = 2 + 6.

Now, the 6 new people that received the email send it to other 3 new people each, so now the number of people that got the email is:

2 + 6 + 6*3

and so on.

We already can see the pattern here, we can write this as:

∑aₙ

Where aₙ = 3*aₙ₋₁

and a₀ = 2.

So with that information, we can find the number of iterations, n, for a given number of people that received the email A.

A = ∑aₙ = (a₀ + a₁ + a₂ + ) = (2 + 2*3 + 2*3*3 + ...)