Problem 15 - Precalculus: Mathematical for Calculus

Question

Determine whether the sequence is arithmetic. If it is arithmetic, find the common difference. $$2.6,4.3,6.0,7.7, \ldots$$

Step-by-Step Solution

Answer

The sequence is arithmetic with a common difference of 1.7.

Step 1: Identify Consecutive Differences

To determine if the sequence is arithmetic, first calculate the differences between consecutive terms. The sequence given is \(2.6, 4.3, 6.0, 7.7, \ldots\). Find the difference between each successive pair of terms:- The difference between \(4.3\) and \(2.6\) is \(4.3 - 2.6 = 1.7\).- The difference between \(6.0\) and \(4.3\) is \(6.0 - 4.3 = 1.7\).- The difference between \(7.7\) and \(6.0\) is \(7.7 - 6.0 = 1.7\).

Step 2: Verify Consistent Differences

Now, check if the differences calculated in Step 1 are consistent. For a sequence to be arithmetic, the differences between consecutive terms must be the same.We found that the difference between each pair of terms is consistently \(1.7\).

Step 3: Conclusion

Since the difference between each pair of consecutive terms in the sequence is \(1.7\) and consistent, the sequence is indeed arithmetic. The common difference is thus \(1.7\).

Key Concepts

Common Difference

In the world of mathematics, an **arithmetic sequence** is defined by its common difference. The common difference is a constant that lies between consecutive terms of the sequence. To find the common difference, you subtract any term from the next term. Here’s how you can think of it:

• Begin by picking any two consecutive numbers in the sequence.
• Subtract the first number from the second.

For example, let’s take our sequence: \(2.6, 4.3, 6.0, 7.7, \ldots\) and calculate the difference between the first term and the second: \(4.3 - 2.6 = 1.7\). You continue this for other consecutive pairs, confirming they all yield the same result. If you keep getting the same value, like \(1.7\) here, then this value is your common difference.This consistency in difference tells us that our sequence changes at a steady rate and is indeed arithmetic. Recognizing this pattern is crucial for analyzing and working with sequences.

Consecutive Terms

A sequence is essentially a list of numbers in a special order. When delving into arithmetic sequences, we focus on **consecutive terms**. These are numbers that follow one after another directly in a sequence. For example, in our sequence: \(2.6, 4.3, 6.0, 7.7, \ldots\), each number is a consecutive term.The distinctive trait of arithmetic sequences is that each consecutive term is built by adding the common difference to the preceding term. Here’s a simple way to visualize it:

• Start with a term, say \(2.6\).
• Add the common difference \(1.7\) to reach the next term: \(4.3\).
• Keep adding \(1.7\) to reach further down the sequence: \(4.3\) to \(6.0\) and so on.

Understanding consecutive terms is like setting up the building blocks of an arithmetic sequence. It's a straightforward yet powerful concept that supports unraveling more complexities in sequence behaviors.

Sequence Analysis

Let's dive deeper into **sequence analysis** to examine whether a set of numbers forms an arithmetic sequence. Analyzing a sequence requires a systematic approach:1. **Identify the sequence elements**: Begin by listing the numbers, like our sequence: \(2.6, 4.3, 6.0, 7.7, \ldots\).2. **Calculate consecutive differences**: Check the differences between each consecutive pair. For instance, take \(4.3 - 2.6\), \(6.0 - 4.3\), and so on. You should find they all share the same value, which confirms consistency.3. **Evaluate the pattern**: If these differences remain unchanged throughout, an arithmetic sequence is confirmed. This shows a linear growth path in the sequence numbers, like stepping up a consistent ladder. Sequence analysis is vital to recognize whether the criteria for certain types of sequences are met. Proper analysis simplifies predictions about future sequence numbers and helps relate sequences to real-world phenomena. It equips learners with a robust mathematical toolset for problem-solving.