On January 1st, Jerome concocts a rumor and tells Beth. On January 2nd, they each tell a new person. By the end of January 2nd, four people have heard the rumor. On each subsequent day, each person who has heard the rumor spreads it to a new person.
Since you didn't exactly finish the question I assume you want to know the number of people that have heard the rumor after some number of days.
To start, let y = the number of people that have heard the rumor and n = the number of days that have passed.
For n = 1; y = 2
n = 2; y = 4
n = 3; y = 8
Looking at this, a pattern jumps out at me. For the first y, we get 2; the second y gets 4 (2*2); the third y gets 8 (2*2*2). From this, we can construct a simple equation to show the number of people that have heard the rumor after n days: y = 2^n.
Hope this helps!