Problem 17 - 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. $$11,17,23,29, \dots$$

Step-by-Step Solution

Answer

Yes, the terms form an arithmetic sequence with a common difference of 6.

Step 1: Check Differences Between Consecutive Terms

To check if the sequence is arithmetic, calculate the difference between consecutive terms. If this difference is the same between all terms, the sequence is arithmetic. Calculate the differences:- Between the 1st and 2nd terms: \(17 - 11 = 6\)- Between the 2nd and 3rd terms: \(23 - 17 = 6\)- Between the 3rd and 4th terms: \(29 - 23 = 6\)Since all these differences are equal, the sequence is arithmetic.

Step 2: Determine the Common Difference

Since the sequence is arithmetic (the differences between consecutive terms are equal), the common difference of the sequence is \(6\).

Key Concepts

Common Difference

In an arithmetic sequence, each term after the first is found by adding a fixed number, called the "common difference," to the preceding term. This common difference is crucial because it is what sets arithmetic sequences apart from other sequences. In our example,

• To find the common difference, subtract the first term, 11, from the second term, 17. The difference is 6.
• Similarly, when you subtract the second term, 17, from the third term, 23, you again get 6.
• This consistency continues when subtracting 23 from 29, giving a difference of 6 once more.

This repetitive difference confirms that the sequence is indeed arithmetic with a common difference of 6. Understanding this concept aids in predicting further terms in the sequence by continually adding this difference to the previous term.

Sequence Analysis

Sequence analysis is about examining the structure of sequences to understand patterns and relationships between terms. For this analysis, it's crucial to determine whether a sequence follows a specific rule or pattern. In arithmetic sequences, the focus is on ensuring that the differences between consecutive terms remain constant. An arithmetic sequence such as our example (11, 17, 23, 29, ...) shows a very clear pattern:

• Examine the differences between each term.
• If the differences are equal, we can conclude the sequence follows an arithmetic pattern.
• This pattern helps in easily identifying future terms by applying the common difference systematically.

To extend the sequence, simply add the common difference to the last known term. Sequence analysis simplifies this process, confirming an arithmetic nature if the differences are consistent.

Precalculus

Precalculus is a course that prepares students for calculus by exploring various types of functions and sequences. Understanding arithmetic sequences fits perfectly into precalculus learning since it enhances comprehension of mathematical progression and series. Using arithmetic sequences as a foundation, students can:

• Learn how to identify patterns and predict terms in sequences.
• Apply algebraic techniques to understand real-world scenarios modeled by linear functions.
• Understand the connection between sequences and series, which is a stepping stone to calculus concepts.
  

For instance, by mastering the rule of an arithmetic sequence, students can better understand how linear equations function, providing a strong basis for tackling calculus problems later on. Precalculus thus acts as a bridge, connecting basic arithmetic sequence knowledge to more advanced mathematical concepts like differentiation and integration.