A new fitness center opened up with 24 clients. Each week they are signing up an average of 5 new members. The business owner wrote the explicit formula for this situation as being A(n) = 24 + (n – 1)(5). Is this correct for this situation? Justify your answer using complete sentences and proper grammar.
The Correct Answer and Explanation is:
The explicit formula provided by the business owner, ( A(n) = 24 + (n – 1)(5) ), is correct, but let’s break it down to fully understand why and provide a clearer formula for this situation.
Step 1: Understanding the Situation
The fitness center initially has 24 clients. Each week, 5 new members are signing up. This means that the total number of clients is increasing by 5 each week, starting from 24 clients.
In mathematical terms:
- The initial number of clients is 24.
- The change in the number of clients each week is 5.
- We are interested in finding out how many clients there are after (n) weeks.
Step 2: Understanding the Formula
The business owner used the formula ( A(n) = 24 + (n – 1)(5) ).
Here’s how we can interpret the formula:
- 24 is the initial number of clients when (n = 1) (the first week).
- (n – 1) represents the number of weeks that have passed since the first week, since the first term, 24, corresponds to (n = 1).
- 5 is the number of new clients signing up each week. Therefore, multiplying (n – 1) by 5 gives the total increase in clients after (n – 1) weeks.
Step 3: Verifying the Formula
Let’s verify the formula with a couple of values for (n):
- For (n = 1):
[ A(1) = 24 + (1 – 1)(5) = 24 + 0 = 24. ]
This matches the initial number of clients. - For (n = 2):
[ A(2) = 24 + (2 – 1)(5) = 24 + 5 = 29. ]
This is correct, as 5 new clients joined, bringing the total to 29.
Thus, the formula correctly calculates the total number of clients for any given week.
Step 4: A Clearer Formula
A more standard explicit formula for this type of situation would be:
[ A(n) = 24 + 5(n – 1). ]
This formula is essentially the same but is often written more clearly. If we distribute the 5 across (n – 1), we get:
[ A(n) = 24 + 5n – 5, ]
which simplifies to:
[ A(n) = 19 + 5n. ]
Both forms are equivalent, but the more direct formula ( A(n) = 19 + 5n ) can be easier to work with for general cases. Therefore, while the business owner’s formula is correct, a clearer version would be ( A(n) = 19 + 5n ).