Problem 17 - Contemporary Precalculus

Question

Show that the sequence is arithmetic and find its common difference. $$\\{3-2 n\\}$$

Step-by-Step Solution

Answer

Q: Prove that the given sequence is an arithmetic sequence and find the common difference. A: The given sequence $$a_n = 3 - 2n$$ is an arithmetic sequence with the common difference $$d = -2$$.

Step 1: Define the sequence and the general term

The given sequence can be represented as: $$a_n = 3 - 2n$$, where $$a_n$$ is the nth term of the sequence.

Step 2: Calculate the first few terms of the sequence

To better understand the sequence, we can calculate its first few terms by substituting the value of $$n$$: For $$n=1$$, $$a_1 = 3 - 2(1) = 1$$ For $$n=2$$, $$a_2 = 3 - 2(2) = -1$$ For $$n=3$$, $$a_3 = 3 - 2(3) = -3$$ Now we have the first three terms of the sequence: $$1, -1, -3$$.

Step 3: Calculate the common difference for consecutive terms

Let's calculate the common difference between consecutive terms: $$d = a_2 - a_1 = -1 - 1 = -2$$ $$d = a_3 - a_2 = -3 - (-1) = -2$$ The common difference is the same between these consecutive terms (-2).

Step 4: Prove that the common difference is constant for all terms

We can now prove that the common difference is constant for all terms: Let's calculate $$a_{n+1} - a_n$$: $$a_{n+1} - a_n = (3 - 2(n+1)) - (3 - 2n)$$ $$a_{n+1} - a_n = (3 - 2n - 2) - (3 - 2n)$$ $$a_{n+1} - a_n = -2$$ This proves that the common difference is constant (-2) for the entire sequence.

Step 5: Conclusion

The sequence $$a_n = 3 - 2n$$ is an arithmetic sequence with a common difference $$d = -2$$.

Key Concepts

Common Difference

The common difference in an arithmetic sequence is the difference between successive terms. It remains the same throughout the sequence. Let’s delve into how this applies to the sequence \(a_n = 3 - 2n\).

By calculating the first few terms, we see:

• \( a_1 = 1 \)
• \( a_2 = -1 \)
• \( a_3 = -3 \)

To find the common difference, take any term and subtract the previous term. For our terms:\( a_2 - a_1 = -1 - 1 = -2 \).
Similarly, \( a_3 - a_2 = -3 - (-1) = -2 \).

The sequence has a constant common difference of \(-2\). This confirms that this sequence is arithmetic.

General Term

The general term in an arithmetic sequence describes any term in the sequence based on its position, \(n\). For our sequence, \(a_n = 3 - 2n\). This formula tells us how to calculate any term of the sequence using its position number, \(n\).

For instance, if \(n=4\), substitute \( n \) in the formula to get \( a_4 = 3 - 2(4) = -5 \).

This general representation helps efficiently find any term without calculating all preceding terms.

Consecutive Terms

Consecutive terms are terms that come one after the other in a sequence. In our example, \(1, -1, -3\) are consecutive terms for \(n=1, 2,\) and \(3\) respectively.

When identifying consecutive terms, you can use them to validate the arithmetic nature of a sequence by checking if the differences between each pair are the same.

As we computed earlier:

• \( a_2 - a_1 = -1 - 1 = -2 \)
• \( a_3 - a_2 = -3 - (-1) = -2 \)

This consistent difference confirms each step moves by \(-2\).

Sequence Analysis

Sequence analysis involves examining the sequence to determine its nature and properties. For our sequence \(a_n = 3 - 2n\), we start by identifying it as arithmetic due to the constant common difference of \(-2\).

We can apply this understanding to forecast terms further along in the sequence, ensuring that every step follows the common pattern: moving backward \(-2\) from each term to the next.

This analysis reveals the linear relationship defining term positions and shows that the sequence will indefinitely continue decreasing, remaining within the realm of arithmetic.