Problem 67 - Contemporary Precalculus

Question

The number of bachelor's degrees (in thousands) awarded in year \(n\) is approximated by the sequence \(\left\\{a_{n}\right\\},\) where \(a_{n}=\) \(25.3 n+1250\) and \(n=0\) corresponds to \(2000 .\) (a) Approximately how many bachelor's degrees were awarded in 2004 and in \(2007 ?\) (b) Approximately how many bachelor's degrees will be awarded between 2004 and 2009 (inclusive)?

Step-by-Step Solution

Answer

Based on the given sequence formula \(\left\\{a_{n}\right\\}\), where \(a_{n} = 25.3n + 1250\), approximately 1,351,200 bachelor's degrees were awarded in 2004 and 1,427,100 bachelor's degrees were awarded in 2007. Additionally, from 2004 to 2009 inclusive, approximately 8,486,700 bachelor's degrees were awarded in total.

Step 1: (Understanding the sequence formula)

The number of bachelor's degrees awarded in year n is given by the sequence \(\left\\{a_{n}\right\\},\) where \(a_{n} = 25.3n + 1250\) and \(n=0\) corresponds to year 2000. To find the number of degrees awarded in a particular year, we can plug the value of n for that year into the sequence formula.

Step 2: (Finding the degrees awarded in 2004 and 2007)

To find the number of degrees awarded in 2004, we need to calculate \(a_{4}\), since 2004 is 4 years after 2000: \(a_4 = 25.3(4) + 1250 = 101.2 + 1250 = 1351.2\). To find the number of degrees awarded in 2007, we need to calculate \(a_{7}\), since 2007 is 7 years after 2000: \(a_7 = 25.3(7) + 1250 = 177.1 + 1250 = 1427.1\).

Step 3: (Answer for part a)

In 2004, approximately 1351.2 thousand (1,351,200) bachelor's degrees were awarded, and in 2007, approximately 1427.1 thousand (1,427,100) bachelor's degrees were awarded.

Step 4: (Finding the total degrees awarded between 2004 and 2009)

To find the total number of bachelor's degrees awarded between 2004 and 2009 (inclusive), we need to add up the values of the sequence for those years: \(a_4 + a_5 + a_6 + a_7 + a_8 + a_9\). #a_{4}# is 1351.2, and #a_{7}# is 1427.1 as calculated above. We now need to calculate the following: \(a_5 = 25.3(5) + 1250 = 126.5 + 1250 = 1376.5\), \(a_6 = 25.3(6) + 1250 = 151.8 + 1250 = 1401.8\), \(a_8 = 25.3(8) + 1250 = 202.4 + 1250 = 1452.4\), and \(a_9 = 25.3(9) + 1250 = 227.7 + 1250 = 1477.7\). Now, we add up all the values: \(1351.2 + 1376.5 + 1401.8 + 1427.1 + 1452.4 + 1477.7 = 8486.7\).

Step 5: (Answer for part b)

Between 2004 and 2009 (inclusive), approximately 8486.7 thousand (8,486,700) bachelor's degrees will be awarded.

Key Concepts

Arithmetic Sequences

Arithmetic sequences are a fundamental concept in algebra that describe a list of numbers where each term is generated by adding a constant to the previous term. This constant is known as the common difference of the sequence. For instance, the sequence 2, 5, 8, 11,... is an arithmetic sequence with a common difference of 3.

In the context of the exercise, the number of bachelor's degrees awarded each year forms an arithmetic sequence. This is because there is a steady increase that can be represented as a fixed amount (the common difference) added each year. The general formula for an arithmetic sequence is given by:
\[a_n = a_1 + (n - 1)d\]
where \(a_n\) is the term we want to find, \(a_1\) is the first term in the sequence, \(n\) is the term's position in the sequence, and \(d\) is the common difference. In the given exercise, the formula for the number of degrees awarded is\[a_n = 25.3n + 1250\]where \(25.3\) represents the common difference, indicating how much the number of degrees increases each year, and \(1250\) is the initial number of degrees awarded in the year 2000.

Practical Application of Arithmetic Sequences

Understanding arithmetic sequences can be incredibly helpful in various scenarios such as predicting trends, projecting sales, budgeting, and planning in businesses. In educational data, it can be used to forecast enrollment numbers, graduation rates, or even plan for the resources needed over time.

Algebraic Formulas

Algebraic formulas are equations that represent relationships between quantities. They are the bedrock of algebra and play a critical role in solving problems that involve unknown variables. An algebraic formula often looks like an equation involving letters (representing variables) and numbers.

The sequence approximation in the exercise, \(a_n = 25.3n + 1250\), is an example of an algebraic formula. It makes it possible to find the number of bachelor's degrees without having to list out the entire sequence. The formula encapsulates the pattern of growth in a simple, time-saving expression. This is key for students and professionals who need to analyze educational trends or predict future outcomes based on historical data.

Algebraic Formulas in Daily Life

Everyday applications of algebraic formulas include calculating interest in finance, converting temperatures between Celsius and Fahrenheit, and determining the speed or distance of objects in physics. Understanding these formulas is vital for solving a wide range of practical problems in science, engineering, economics, and beyond.

Educational Data Analysis

Educational data analysis is the process of collecting, processing, and interpreting data to make informed decisions in an educational context. It involves a variety of techniques and tools aimed at improving student outcomes and optimizing educational strategies.

In the given exercise, analyzing the sequence of bachelor's degrees awarded over time is a simple form of educational data analysis. By applying the algebraic formula for the arithmetic sequence, one can deduce not only the number of degrees awarded in a single year but also the total degrees awarded over a range of years. Such analysis can aid in strategic planning, policy formulation, forecasting future needs, and assessing the impact of educational programs.

The Role of Analysis in Educational Planning

Educational institutions rely on data analysis to track progress, identify areas that need improvement, and allocate resources efficiently. It supports decisions such as faculty hiring, curriculum development, and student support services. For students learning about educational data analysis, it's crucial to understand how to interpret data sets and trends to help shape the educational landscape for the better.