Problem 62 - Algebra and Trigonometry

Question

The bar graph shows changes in the percentage of college graduates for Americans ages 25 and older from 1990 to 2010 . Involve developing arithmetic sequences that model the data. Percentage of College Graduates for Americans Ages 25 and Older (Graph cannot copy) In \(1990,24.4 \%\) of American men ages 25 and older had graduated from college. On average, this percentage has increased by approximately 0.3 each year. A. Write a formula for the \(n\) th term of the arithmetic sequence that models the percentage of American men ages 25 and older who had graduated from college \(n\) years after 1989 B. Use the model from part (a) to project the percentage of American men ages 25 and older who will be college graduates by 2019

Step-by-Step Solution

Answer

The formula for the nth term of the arithmetic sequence modeling the percentage of American men ages 25 and older who had graduated from college n years after 1989 is \(a_{n} = 24.4 + (n-1)0.3\). Therefore, it is projected that 33.1% of American men ages 25 and older will be college graduates by 2019.

Step 1: Understand the data

The given data indicate that in 1990, 24.4% of American men ages 25 and older had graduated from college. Also, each year, this percentage has increased by approximately 0.3. It means 1990 is the starting point and observed as '0' in the sequence. Therefore, the first term (a) of the sequence is 24.4 and the common difference (d) of the sequence is 0.3.

Step 2: Formulate the arithmetic sequence

An arithmetic sequence can be represented by \(a_{n} = a + (n-1)d\), where \(a_{n}\) is the nth term of the sequence, 'a' is the first term and 'd' is the common difference. Substituting the values from the data (a=24.4, d=0.3) into the formula yields \(a_{n} = 24.4 + (n-1)0.3\).

Step 3: Project future data

In 2019, n equals to 30 (from 1989 to 2019). To determine the percentage of American men ages 25 and older who will be college graduates by 2019, substitute 30 for 'n' in the arithmetic sequence developed in the previous step and calculate. The solution for \(a_{n} = 24.4 + (30-1)0.3\) gives 33.1.

Key Concepts

College Graduation Rates

Graduation rates give us insights into the educational achievements of a population. In particular, when we talk about college graduation rates, we refer to the percentage of people who have successfully completed a college degree within a certain age group. In this case, we're focusing on American men aged 25 and older.

Over time, these rates can reflect various socio-economic factors, such as access to higher education and societal emphasis on the importance of a college degree. Understanding these rates allows us to track educational progress and pinpoint areas that need improvement. In our exercise, these rates are modeled as an arithmetic sequence, illustrating steady annual growth over two decades.

Sequence Formula

A sequence formula is crucial in calculating the terms of a sequence based on its position. For arithmetic sequences, the formula expresses each term as a function of its position (or sequence number) and the constants defining the sequence.

Here, our arithmetic sequence is defined by the formula:

• \(a_n = a + (n-1)d\)

Where:

• \(a_n\) is the \(n\)th term of the sequence,
• \(a\) is the initial term (or starting value),
• \(n\) is the term number we're interested in,
• \(d\) is the common difference.

This formula helps us predict graduation rates in any given year, highlighting the power of arithmetic sequences to model real-world data.

Common Difference

The common difference is a key part of arithmetic sequences. It represents the consistent amount added to each term to get the next term. In the context of college graduation rates, it's the average annual increase in the percentage of graduates.

For our given sequence, the common difference is 0.3. This means that each year, the graduation rate increases by approximately 0.3%. By knowing the common difference, we can easily calculate future terms using the sequence formula. It brings predictability and a straightforward approach to forecasting trends.

Projection in Sequences

Projection in sequences involves predicting future terms of a sequence using established patterns. This is particularly useful in planning and strategy, whether it be in education, sales forecasts, or population studies.

To project the graduation rate for 2019, we determine how many years have passed since the initial term from 1989. Here, 30 years had passed by 2019. Inserting this into our sequence formula \(a_n = 24.4 + (30-1) \times 0.3\), we find that the predicted graduation rate in 2019 is \(33.1\%\).

Such projections help stakeholders make informed decisions regarding educational policies and resource allocation based on expected future scenarios.