Question

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100\)

Short answer

Answer: The limit of the sequence does not exist.

Step-by-step solution

Determine the first five terms of the sequence

We are given that the base value for the CPI in 1984 is \(c_0 = 100\). The CPI increases by 3% each year, so we can find the subsequent terms by multiplying the previous term by \(1 + \frac{3}{100}\). \(c_1 = c_0 \cdot \left(1 + \frac{3}{100}\right) = \boxed{103}\). \(c_2 = c_1 \cdot \left(1 + \frac{3}{100}\right) = 103\cdot \left(1 + \frac{3}{100}\right) = \boxed{106.09}\). \(c_3 = c_2 \cdot \left(1 + \frac{3}{100}\right) = 106.09\cdot \left(1 + \frac{3}{100}\right) = \boxed{109.2727}\). \(c_4 = c_3 \cdot \left(1 + \frac{3}{100}\right) = 109.2727\cdot \left(1 + \frac{3}{100}\right) = \boxed{112.55078}\). \(c_5 = c_4 \cdot \left(1 + \frac{3}{100}\right) = 112.55078\cdot \left(1 + \frac{3}{100}\right) = \boxed{115.9273034}\). The first five terms of the sequence are: 100, 103, 106.09, 109.2727, and 112.55078.

Find an explicit formula for the terms of the sequence

Since the CPI increases by 3% each year, the sequence follows a geometric progression. The general form for a geometric sequence is given by \(c_n = c_0 \cdot r^n\), where \(c_n\) is the nth term, \(c_0\) is the initial term, \(r\) is the common ratio, and \(n\) is the number of years. In this case, \(c_0 = 100\) and \(r = 1 + \frac{3}{100}\). So, the explicit formula for the terms of the sequence is: \(c_n = 100 \cdot \left(1 + \frac{3}{100}\right)^n = 100 \cdot \left(\frac{103}{100}\right)^n\).

Find a recurrence relation that generates the sequence

To find a recurrence relation for this sequence, we can express a term \(c_n\) in terms of its previous term \(c_{n-1}\). Since each term is obtained by multiplying the previous term by \(1 + \frac{3}{100}\), we have the recurrence relation: \(c_n = c_{n-1} \cdot \left(1 + \frac{3}{100}\right)\), with \(c_0 = 100\).

Estimate the limit of the sequence

To estimate the limit of the sequence, we will use a calculator or a graphing utility. However, we can notice that since the common ratio \(r = 1 + \frac{3}{100} > 1\), the sequence is a geometric progression with a common ratio greater than 1, which means it will continue to grow indefinitely. Therefore, the limit of the sequence does not exist.

Key concepts

Consumer Price Index (CPI)

The Consumer Price Index (CPI) is a critical economic indicator that measures the change in the cost of a standard basket of goods and services purchased by typical households. The CPI is used to calculate inflation, which reflects the rate at which prices are rising. In educational exercises, such as the one given, the CPI is often used to illustrate real-world applications of mathematical sequences.

Understanding the CPI is useful for understanding economic health, making personal financial decisions, and for students, it provides a tangible context to apply mathematical concepts. The base value, set at 100 in the year 1984 for this exercise, serves as a reference point, allowing us to calculate the percentage increase in prices over time.

Explicit Formula for a Sequence

An explicit formula for a sequence provides a way to calculate the value of any term in the sequence without needing to know the previous terms. It precisely defines the nth term as a function of n, which is the term's position in the sequence. For a geometric sequence, like the one representing the CPI in our exercise, the explicit formula takes the form \(c_n = c_0 \times r^n\), where \(c_0\) is the first term, r is the common ratio, and n is the term index.

For students, the explicit formula is a powerful tool as it facilitates quick calculations of terms deep into the sequence and aids in understanding long-term trends without laborious step-by-step computations.

Recurrence Relation

A recurrence relation is a way of defining the terms of a sequence using the previous terms. It's a formula that recursively tells us how to get from one or more previous terms to the next. In the context of our CPI sequence, we have a simple first-order linear recurrence relation \(c_n = c_{n-1} \times (1 + \frac{3}{100})\), which states that each year's CPI (\(c_n\)) is obtained by multiplying the previous year's CPI (\(c_{n-1}\)) by the growth factor (1 + \(\frac{3}{100}\)).

This relation emphasizes the dependence of each term on its predecessor and is particularly useful when we are looking at the sequence term-by-term or when an explicit formula is unavailable or difficult to derive.

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as the index n increases indefinitely. For geometric sequences, if the common ratio is between -1 and 1, the limit is found by the term eventually getting closer and closer to zero. However, if the common ratio is greater than 1 or less than -1, the sequence will either grow indefinitely or oscillate without approaching a single value, meaning the limit does not exist.

In our given problem, since the common ratio is greater than 1 (\(r = 1 + \frac{3}{100} > 1\)), the CPI will increase infinitely over time. Thus, the sequence has no limit, providing a model that reflects unbounded inflation over the years. This concept is important not just mathematically but also economically, as it reflects scenarios of persistent inflationary periods.