Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether or not the sequence is arithmetic. If it is, find the common difference. $$3.7,3.1,2.5,1.9,1.3, \ldots$$

Step-by-Step Solution

Answer

Yes, the sequence is arithmetic and the common difference is -0.6.

Step 1: Confirm If Sequence is Arithmetic

Subtract the second term from the first term, the third term from the second term, the fourth term from the third term, and so on, to ensure the differences are constant. If they are, then it is confirmed that the sequence is arithmetic. For this sequence, \(3.1 - 3.7 = -0.6\), \(2.5 - 3.1 = -0.6\), \(1.9 - 2.5 = -0.6\), \(1.3 - 1.9 = -0.6\). Therefore, all differences are equal to -0.6, hence the sequence is arithmetic.

Step 2: Find the Common Difference

Once it's known that the sequence is arithmetic, the common difference can be ascertained. This is simply the constant difference between every two consecutive terms in the sequence. It's clear that the common difference here is -0.6, as calculated earlier.

Key Concepts

Understanding the Common Difference in Arithmetic Sequences

In an arithmetic sequence, a common difference is a consistent number added to or subtracted from each term to get to the next. This is a key characteristic that defines whether a sequence is arithmetic or not. If you don't have a constant difference, it's not an arithmetic sequence.

To find this common difference:

• Subtract the first term from the second term.
• Repeat this for subsequent terms.

If these differences remain the same, you have confirmed the common difference.

For example, in the sequence \(3.7, 3.1, 2.5, 1.9, 1.3, \ldots\) we see that the difference \(-0.6\) consistently applies, confirming the sequence as arithmetic.By clearly understanding the role of the common difference, you can easily determine and classify arithmetic sequences.

Sequence Determination Process

Determining whether a sequence is arithmetic involves checking for a consistent pattern between terms. This methodical approach allows us to confidently categorize the sequence type.

Here's how to determine if a sequence like \(3.7, 3.1, 2.5, 1.9, 1.3, \ldots\) is arithmetic:

• Calculate the difference between consecutive terms.
• Ensure that this difference is constant throughout the sequence.

If each step yields the same result, you can affirm that the sequence is arithmetic.

Our example sequence had a consistent difference of \(-0.6\), verifying it as arithmetic. Determining a sequence isn't just about plugging numbers; it requires observation and logical reasoning.

Application of Mathematical Reasoning

Mathematical reasoning is crucial in identifying and confirming arithmetic sequences. It involves logical thinking and deduction. By applying reasoning, you ensure accuracy in the analysis.

Here's how mathematical reasoning is applied in our example:

• Observe the sequence to see an apparent pattern between terms.
• Analyze the sequence by calculating differences to verify this pattern.
• Deduce, from consistent differences, that the sequence is arithmetic.

Using reasoning not only helps in confirming if a sequence is arithmetic, but also deepens your understanding of mathematical patterns. This critical thinking skill is useful beyond specific arithmetic problems.

Fostering a habit of reasoning ensures that your approach to any sequence problem is both systematic and informed.