Problem 20 - Precalculus: Mathematics for Calculus

Question

The first four terms of a sequence are given. Can these terms be the terms of an arithmetic sequence? If so, find the common difference. $$100,68,36,4, \dots$$

Step-by-Step Solution

Answer

Yes, it is an arithmetic sequence with a common difference of -32.

Step 1: Identify the Given Terms

The sequence given consists of the first four terms: 100, 68, 36, and 4.

Step 2: Check for a Common Difference

To determine if the sequence is arithmetic, calculate the difference between consecutive terms. An arithmetic sequence has a consistent difference between each term.

Step 3: Calculate the First Difference

Subtract the second term from the first: \[ d_1 = 68 - 100 = -32 \]

Step 4: Calculate the Second Difference

Subtract the third term from the second: \[ d_2 = 36 - 68 = -32 \]

Step 5: Calculate the Third Difference

Subtract the fourth term from the third: \[ d_3 = 4 - 36 = -32 \]

Step 6: Verify the Consistent Difference

Since all differences \(d_1 = d_2 = d_3 = -32\), the sequence is arithmetic with a common difference of \(-32\).

Key Concepts

Common Difference

An arithmetic sequence is a type of sequence where consecutive terms have a constant difference. This difference is known as the "common difference." It is a key feature that distinguishes an arithmetic sequence from other types of sequences. In general, if you have a sequence of numbers, you can check whether it is arithmetic by subtracting each term from its following term. If the difference remains the same throughout, it indicates a common difference.

• If the sequence is increasing, the common difference will be positive.
• If the sequence is decreasing, the common difference will be negative.
• For example, in the sequence 100, 68, 36, 4, each term decreases by 32 points, resulting in a common difference of \(-32\).

Understanding the common difference helps to easily predict future terms in the sequence and identify the nature of the sequence.

Sequence and Series

A sequence is an ordered list of numbers following a particular pattern or rule. A series, on the other hand, is the sum of the terms of a sequence. Sequences can be finite or infinite and are classified into different types based on their rules.

• In arithmetic sequences, the rule involves adding or subtracting the common difference to get the next term.
• For example, using the sequence 100, 68, 36, 4, the next term can be found by subtracting \(-32\) from 4, resulting in \(-28\).
• This predictable nature of sequences allows for easy calculation of unknown terms.

The concept of sequences and series is fundamental in mathematics, providing foundational understanding for more advanced studies. Recognizing patterns in sequences helps in developing problem-solving skills and analytical thinking.

Precalculus

Precalculus serves as the bridge between algebra and calculus, providing necessary tools and concepts to prepare students for calculus studies. One of the key topics in precalculus is understanding various types of sequences, including arithmetic sequences.

• Precalculus emphasizes comprehending how sequences progress and how series are sum of sequences.
• The ability to identify and work with sequences can simplify complex calculus problems later on.
• Understanding arithmetic sequences, like the one given with terms 100, 68, 36, and 4, helps in grasping the idea of linear patterns and functions.

This foundation supports future calculus concepts, where sequences often relate to limits, functions, and deriving more complex mathematical models. Precalculus is thus an essential step in a student's mathematical journey, paving the way for success in calculus and beyond.