Lucas’s friend Isabel is performing in a singing competition. The winner of the competition will be determined by
call-in votes. To help Isabel earn votes, Lucas sends text messages to 8 of his friends. He then asks each of his friends to send texts to 8 more friends, and asks his friends to ask each of their friends to send 8 texts. He is hoping for this pattern to continue and that many people will receive text messages telling them to vote for Isabel. Each set of text messages is a round of text messages, where Lucas’s messages are the first round, Lucas’s friends messages are the second round, and so on.
What would a diagram look like modeling the first few rounds of the situation?
If you know the number of texts sent in round x-1, how many texts will be sent in round x?
How many texts will be sent in round x? In other words, what is the explicit definition (algebraic expression) of the number of texts sent in any round, x?