Problem 80 - College Algebra with Modeling and Visualization

Question

Given the terms of a finite sequence, classify it as arithmetic, geometric, or neither. $$5.75,5.5,5.25,5,4.75,4.5$$

Step-by-Step Solution

Answer

The sequence is arithmetic.

Step 1: Identify the type of sequence

To classify the sequence, either arithmetic or geometric, we need to verify if there is a common difference between consecutive terms (arithmetic) or a common ratio (geometric). The given sequence is: \(5.75, 5.5, 5.25, 5, 4.75, 4.5\).

Step 2: Determine Difference Between Terms

Calculate the difference between consecutive terms: \(5.5 - 5.75 = -0.25\), \(5.25 - 5.5 = -0.25\), \(5 - 5.25 = -0.25\), \(4.75 - 5 = -0.25\), and \(4.5 - 4.75 = -0.25\). The difference is consistent across terms.

Step 3: Confirm Arithmetic Sequence

Since the difference \(-0.25\) between each consecutive term is constant, the sequence is arithmetic. There is no need to check for a common ratio as arithmetic has been confirmed.

Key Concepts

finite sequence

A finite sequence is a set of numbers that has a limited number of terms. In this type of sequence, you can count the number of elements from start to finish, meaning it doesn’t go on indefinitely. Finite sequences are often represented with a list of numbers ending at a specific point. This is different from an infinite sequence, which continues without end.

• Finite sequences have a definite, countable number of terms.
• Examples include a list of attendance numbers, steps to complete a task, or the finite sequence given in the exercise: 5.75, 5.5, 5.25, 5, 4.75, 4.5.
• They are useful in various mathematical contexts to identify patterns or trends.

Understanding whether a sequence is finite or infinite helps in determining how to approach and solve related problems. This distinction can impact how you analyze and predict the sequence's behavior.

common difference

The concept of a common difference is central to recognizing an arithmetic sequence. A common difference is the consistent amount you add or subtract to get from one term to the next in a sequence. In the given exercise, for instance, each term in the sequence decreases by 0.25.

• To find the common difference, subtract any term from the next term in the sequence.
• If the difference is the same for all consecutive terms, the sequence is arithmetic.

Identifying the common difference is like finding the key that unlocks the pattern of the sequence. Recognizing this allows you to both confirm the type of sequence and to predict future terms. This understanding simplifies solving problems involving arithmetic sequences.

geometric sequence

A geometric sequence is another type of pattern found in sequences. In a geometric sequence, each term is derived by multiplying the previous term by a constant known as the common ratio. This is different from an arithmetic sequence where you add or subtract a common difference.

• This type of sequence follows an exponential growth pattern.
• An example is the sequence 3, 6, 12, 24, which multiplies by a common ratio of 2.
• To find if a sequence is geometric, divide consecutive terms to see if the ratio remains the same.

Unlike the exercise sequence given, geometric sequences grow multiplicatively, and knowing how to identify them can be invaluable for solving problems that involve exponential relationships. This understanding helps you to quickly categorize and solve sequence-based questions.