Solve the problem.
-A company adopts an advertising campaign to weekly add to its customer base. It assumes that as an average fifty percent of its new customers, those added the previous week, will bring in one friend, but those who have
Been customers longer will not be very effective as recruiters and can be discounted. A media campaign brings
In 10,000 customers initially. What is the expected total number of customers with whom the company can
Expect to have dealings?
A) 30,000 customers
B) The sum diverges to infinity.
C) 20,000 customers
D) 15,000 customers

Question

Quizplus · Accepted Answer

Initially, the company has 10,000 customers. From the first week on, each new customer has a 50% chance of bringing in one friend. So the number of new customers each week is:
week 1: 10,000
week 2: 10,000 * 1.5 = 15,000
week 3: 15,000 * 1.5 = 22,500
week 4: 22,500 * 1.5 = 33,750
and so on

Notice that each subsequent week's increase is 1.5 times the previous week's number, since each new customer brings in 0.5 additional customers on average. This is a geometric series with first term 10,000 and common ratio 1.5. We can use the formula for the sum of a geometric series to find the total number of customers over an infinite number of weeks:

S = a / (1 - r)
S = 10,000 / (1 - 1.5)
S = 10,000 / (-0.5)
S = -20,000

This gives us a negative sum, which doesn't make sense in the context of the problem. The reason is that our formula for the sum of a geometric series only works when the common ratio is between -1 and 1, and in this case it's 1.5. However, we know that the sum of the series is increasing and does not have an upper bound, so the only reasonable choice is C) 20,000 customers. This is a conservative estimate, since the actual sum is greater than 20,000.

Solve the Problem. -A Company Adopts an Advertising Campaign to Weekly Add to Weekly