Problem 21 - Algebra for College Students

Question

$$ 2,6,10,14,18, \ldots \quad 4 n-2 $$

Step-by-Step Solution

Answer

The given sequence is an arithmetic sequence defined by the formula \(a_n = 4n - 2\).

Step 1: Recognize the Pattern

Observe the sequence: 2, 6, 10, 14, 18. Notice how each term increases by a consistent amount.

Step 2: Determine the Common Difference

Subtract the first term from the second term: \(6 - 2 = 4\). Do the same for the next terms: \(10 - 6 = 4\), \(14 - 10 = 4\). The common difference is 4.

Step 3: Identify the General Formula

The sequence is arithmetic with the first term \(a_1\) being 2, and the common difference \(d = 4\). An arithmetic sequence can be expressed as \(a_n = a_1 + (n-1) imes d\).

Step 4: Write the Formula for the nth Term

Using the values of \(a_1 = 2\) and \(d = 4\), substitute these into the formula: \(a_n = 2 + (n-1) imes 4\). Simplify to get \(a_n = 2 + 4n - 4 = 4n - 2\).

Step 5: Verify the Formula

Use the obtained formula to verify it generates the correct terms: For \(n=1\), \(4(1) - 2 = 2\); for \(n=2\), \(4(2) - 2 = 6\); for \(n=3\), \(4(3) - 2 = 10\); and so on. The formula successfully generates the sequence.

Key Concepts

Common Difference

In an arithmetic sequence, one of the most vital components is the "common difference." This is the constant amount by which each term increases compared to the previous term. To find the common difference in any sequence, you simply subtract any term from the term that follows it. Considering the sequence from our example: 2, 6, 10, 14, and 18, each number increases by the same constant, which is 4.

• Subtract 2 from 6 to get 4.
• Subtract 6 from 10 to confirm the difference is again 4.
• The same process applies to 10 and 14, and 14 and 18.

The common difference (d) being 4 is crucial because it helps construct the sequence's general formula and determine successive terms effectively.

General Formula

The general formula of an arithmetic sequence allows us to find any term in the sequence without having to list every preceding term. The formula is derived from the knowledge of the first term and the common difference. It looks like this:
\[ a_n = a_1 + (n-1) \times d \]where:

• \(a_n\) is the nth term.
• \(a_1\) is the first term.
• \(d\) is the common difference.
• \(n\) is the position of the term in the sequence.

In our example, the first term \(a_1\) is 2 and the common difference \(d\) is 4. By plugging these values into the formula, we achieve the general formula \(a_n = 2 + (n-1) \times 4\), simplifying to \(a_n = 4n - 2\). This allows any term in the sequence to be calculated find when its position \(n\) is known.

Nth Term

The nth term of an arithmetic sequence is the particular term located at the n-th position in the sequence. By using the general formula, you can efficiently find the nth term without manually progressing through the entire sequence. The formula we have derived simplifies to:
\[ a_n = 4n - 2 \]This formula essentially lets you determine the specific value of any term based solely on its sequence position \(n\).
For example:

• If \(n = 1\), then \(a_1 = 4(1) - 2 = 2\).
• If \(n = 2\), then \(a_2 = 4(2) - 2 = 6\).
• If \(n = 3\), then \(a_3 = 4(3) - 2 = 10\).
• And so forth.

This method is particularly useful as it allows you to target any specific term in the future sequence easily, illustrating the power of understanding the nth term in algebraic expressions.