How many people would know the fire-safety basics after 10-months? Problem Solving?

The volunteer firefighters were frustrated with their efforts at basic fire prevention training. It seemed that no matter how many fire - safety meetings they set up, hardly anybody ever paid attention. They devised a plan. They decided that each of them (there were eight firefighters) would teach two other people the fire-safety basics. At that point, the teacher would retire but each student would then teach two others. Those people, in turn, would teach two others. The whole thing would be mandated by the city council, and each person would have a month to fulfill his or her teaching requirement. The firefighters taught the first group of people in the first month. Under this plan, how many people would know the fire-safety basics after 10 months?

Public Comments

1. 12,288?
2. At the beginning (t = 0 months), the number of people N who knew fire-safety basics were the eight firefighters. So, N(0) = 8 After the first month (t = 1), 16 more people knew fire-safety basics. So, N(1) = 8 + 16 = 24 After the second month, 32 more people knew fire-safety basics. So, N(2) = 8 + 16 + 32 = 56 Note that all the terms in the sum are powers of two. That is, N(2) = 2^3 + 2^4 + 2^5 So, after the third month, N(3) = 2^3 + 2^4 + 2^5 + 2^6, ... and N(4) = 2^3 + 2^4 + 2^5 + 2^6 + 2^7 In general, N(t) = 2^3 + 2^4 + ... + 2^(t + 3) = 8 (2^0 + 2^1 + ... + 2^t) Also, N(t + 1) - N(t) = [ 2^3 + 2^4 + ... + 2^(t + 3) + 2^(t + 4) ] - [ 2^3 + 2^4 + ... + 2^(t + 3) ] = 2^(t + 4) = (16) (2^t) So, N(t) = (16) (2^t) - 8 And, N(10) = (16) (2^10) - 8 = 16,376