A track coach wants to contact all the households of the students on her team to let them know about a canceled practice. She calls three households. Every household that gets a call then calls two more households (in a phone tree) until the entire team has been notified. Write a geometric series to model how many households there are if it takes four rounds of calling to notify everyone.
The number of households notified in each round can be represented by a geometric sequence with a common ratio of 2.
Let's denote the number of households contacted in the first round as "a", the number of households contacted in the second round as "ar", the number of households contacted in the third round as "ar^2", and the number of households contacted in the fourth round as "ar^3".
Given that it takes four rounds to notify the entire team, we have:
a + ar + ar^2 + ar^3 = total number of households on the team
Since the number of households doubles in each round, we can express the number of households reached in each round as:
a = 3
ar = 3*2^1 = 6
ar^2 = 3*2^2 = 12
ar^3 = 3*2^3 = 24
Therefore, the total number of households on the team after four rounds of calling would be:
3 + 6 + 12 + 24 = 45
Thus, a geometric series to model the number of households if it takes four rounds of calling to notify everyone is 3 + 6 + 12 + 24 = 45.