Problem 25 - 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. $$2.6,4.3,6.0,7.7, \dots$$

Step-by-Step Solution

Answer

Yes, it's an arithmetic sequence with a common difference of 1.7.

Step 1: Recognize Arithmetic Sequence Definition

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant is known as the common difference.

Step 2: Calculate First Difference

To determine the common difference, subtract the first term from the second term: \[ 4.3 - 2.6 = 1.7 \]

Step 3: Calculate Second Difference

Now subtract the second term from the third term: \[ 6.0 - 4.3 = 1.7 \]

Step 4: Calculate Third Difference

Finally, subtract the third term from the fourth term:\[ 7.7 - 6.0 = 1.7 \]

Step 5: Verify Consistency of Differences

All calculated differences are the same (1.7), which shows a constant difference between consecutive terms.

Step 6: Confirm Arithmetic Sequence

Since the differences between each consecutive term are equal, the sequence is indeed an arithmetic sequence.

Step 7: Conclusion on Common Difference

The common difference for the arithmetic sequence is 1.7.

Key Concepts

Understanding the Common Difference

In arithmetic sequences, the term 'common difference' holds a key significance. It refers to the fixed amount added (or subtracted) between each pair of consecutive terms. Finding this difference is crucial for establishing whether a sequence is arithmetic. For instance, let's look at the mention of the sequence: 2.6, 4.3, 6.0, and 7.7. By taking the difference between each consecutive pair of numbers, we can determine if the sequence follows an arithmetic pattern. To do this, subtract the first term from the second term, the second term from the third, and so on. In this exercise, each subtraction results in 1.7:

• First Difference: 4.3 - 2.6 = 1.7
• Second Difference: 6.0 - 4.3 = 1.7
• Third Difference: 7.7 - 6.0 = 1.7

Since this difference is consistent throughout, it becomes our common difference. Recognizing and evaluating this concept can simplify the process of classifying sequences.

Succession in Sequences

The succession in sequences refers to the order and the continuity of terms within a sequence. In arithmetic sequences, each term relies on a definite pattern dictated by the common difference. This predictable nature makes arithmetic sequences easy to comprehend. To depict how terms succeed one another, notice that each term after the first is found by adding the common difference to the preceding term. For the sequence we're examining, this means:

• Start with the initial term (2.6)
• Add the common difference (1.7) to get the next term: 2.6 + 1.7 = 4.3
• Continue this process: 4.3 + 1.7 = 6.0, and 6.0 + 1.7 = 7.7

Understanding this pattern is crucial for predicting future terms. It allows students to extend the sequence or find any specific term in sequence without direct calculation of all preceding terms.

Sequence Verification

Sequence verification is a process to confirm that a given sequence abides by its defining characteristics. For arithmetic sequences, verification is tied to validating the common difference. To verify an arithmetic sequence, simply check that the difference between consecutive terms remains constant. This is precisely what was done in our exercise.

• Calculate differences between each consecutive pair of terms.
• If all differences are consistent, the sequence is arithmetic.

In our case, all differences were the same, 1.7, confirming the sequence is arithmetic. Verification solidifies our understanding of a sequence's structure, ensuring it meets the necessary criteria to be classified appropriately. It illustrates logical progression and reinforces our initial steps of identifying the sequence type.