Problem 48 - Beginning and Intermediate Algebra

Question

In January \(2008,\) approximately 1000 customers at a grocery store used the self-checkout lane. The owners predict that number will increase by \(20 \%\) per month for the next year. a) Find the general term, \(a_{m},\) of the geometric sequence that models the number of customers expected to use the self-checkout lane each month for the next year. b) Predict how many people will use the self-checkout lane in September 2008 . Round to the nearest whole number.

Step-by-Step Solution

Answer

a) The general term of the geometric sequence that models the number of customers expected to use the self-checkout lane each month for the next year is \(a_m = 1000 \cdot (1.2)^{m-1}\). b) In September 2008, approximately 3758 customers are predicted to use the self-checkout lane.

Step 1: Identify the first term#a_1# and the common ratio#r#.

The first term of the geometric sequence, #a_1#, represents the number of customers using the self-checkout lane in January 2008, which is 1000. The common ratio #r# is the rate at which the number of customers is predicted to increase each month, which is 20%. Convert the percentage increase to a decimal by dividing by 100: \(r = 1 + \frac{20}{100} = 1 + 0.2 = 1.2\)

Step 2: Write the general term formula for a geometric sequence.

The general term of a geometric sequence is given by the formula: \(a_m = a_1 \cdot r^{m-1}\) where, \(a_m\) = the term representing the number of customers in the mth month \(a_1\) = the first term (number of customers in January 2008) \(r\) = the common ratio (1.2)

Step 3: Substitute the values for #a_1# and #r# to find the general term formula for this problem.

Using the given values for \(a_1\) and \(r\), we can write the general term formula for our problem as follows: \(a_m = 1000 \cdot (1.2)^{m-1}\) Now we have our general term formula, \(a_m = 1000 \cdot (1.2)^{m-1}\), which models the number of customers expected to use the self-checkout lane each month for the next year. #b) Predicting the number of customers using the self-checkout lane in September 2008#

Step 1: Identify the month number for September 2008.

Since January 2008 is considered the first month (m=1), we need to find the month number for September 2008. January - September is a difference of 8 months, so September 2008 corresponds to the 9th month (m=9).

Step 2: Substitute the month number (m=9) into the general term formula.

Using our general term formula from part (a) (\(a_m = 1000 \cdot (1.2)^{m-1}\)), substitute the value of m=9 (September 2008) to find the number of customers expected to use the self-checkout lane in September 2008. \(a_9 = 1000 \cdot (1.2)^{9-1}\)

Step 3: Calculate the value of the 9th term in the sequence and round to the nearest whole number.

Find the value of the 9th term (\(a_9\)) by calculating: \(a_9 = 1000 \cdot (1.2)^8 \approx 3758.474\) Round the result to the nearest whole number: \(a_9 \approx 3758\) So the prediction for the number of people using the self-checkout lane in September 2008 is approximately 3758 customers.

Key Concepts

Geometric Sequence Formula

Understanding geometric sequences helps in modeling situations where there is a constant growth rate. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
To find any term in such a sequence, we use the geometric sequence formula:
\[a_m = a_1 imes r^{m-1}\]
where:

• \(a_m\) is the term we're finding,
• \(a_1\) is the first term in the sequence,
• \(r\) is the common ratio,
• \(m\) is the term number we're trying to find.

In our exercise, the first term \(a_1 = 1000\) represents the number of customers, and the common ratio \(r = 1.2\), which reflects a 20% growth each month. By substituting these values into the formula, we create a model for customer growth over time.

Self-Checkout Lane Prediction

The problem of predicting usage of a self-checkout lane involves understanding how an initial number of users grows over time. This scenario is modeled through a geometric sequence, as it exhibits exponential growth.
In our example, the grocery store starts with 1000 customers using self-checkout lanes, and this number is expected to increase by 20% monthly. By applying the geometric sequence formula, we can predict future usage.
For instance, to predict the number of users by September 2008, which is the 9th month (\(m = 9\), we substitute into our formula:
\[a_9 = 1000 imes (1.2)^{9-1}\]
This calculation gives us the number of customers expected to use the self-checkout and shows the practical application of geometric sequences in making business predictions. This helps businesses plan for resources and optimize customer experience.

Customer Growth Rate

The concept of customer growth rate is central to understanding changes in customer behaviors over time. A growth rate signifies the percentage increase in the number of customers over a specific period.
In our problem, a 20% monthly growth means each month the customer base is 20% higher than the previous month. In mathematical terms, this is represented by the common ratio \(r = 1.2\) in our geometric sequence.
Such growth rates are vital for businesses to predict future demand and adjust their strategies accordingly. By calculating accurate predictions, businesses can effectively manage inventory, staffing, and overall customer service.
It also allows businesses to make informed decisions about marketing strategies and expansions, proving the importance of understanding and applying these mathematical concepts.