Problem 6 - Algebra for College Students

Question

List all terms of each finite sequence. \(a_{n}=2 n-1\) for \(1 \leq n \leq 4\)

Step-by-Step Solution

Answer

The terms are 1, 3, 5, and 7.

Step 1: Identify the Sequence Formula

The given sequence formula is \(a_{n} = 2n - 1\). This formula will be used to calculate each term in the sequence by substituting values of \(n\).

Step 2: Substitute \(n = 1\)

Substitute \(n = 1\) into the formula: \(a_{1} = 2(1) - 1 = 1\). So the first term is 1.

Step 3: Substitute \(n = 2\)

Substitute \(n = 2\) into the formula: \(a_{2} = 2(2) - 1 = 3\). So the second term is 3.

Step 4: Substitute \(n = 3\)

Substitute \(n = 3\) into the formula: \(a_{3} = 2(3) - 1 = 5\). So the third term is 5.

Step 5: Substitute \(n = 4\)

Substitute \(n = 4\) into the formula: \(a_{4} = 2(4) - 1 = 7\). So the fourth term is 7.

Step 6: List All Terms

The terms of the sequence for \(1 \leq n \leq 4\) are: 1, 3, 5, 7.

Key Concepts

Sequence Formula

In mathematics, a sequence formula can generate terms of a sequence. Here, the given sequence formula is essential to understanding the solution. The provided formula is: \(a_n = 2n - 1\). It expresses each term of the sequence as a function of the position, \(n\), within the sequence. By plugging in different values for \(n\), we can calculate specific terms.

Key characteristics of the sequence formula:

• \(a_n\) represents the general term of the sequence.
• \(n\) is a position variable, beginning at 1 and increasing by integers.
• The term is found by substituting \(n\) into the formula.

Understanding the sequence formula helps in mastering sequences in mathematics.

Substitution

Substitution is a fundamental step in applying the sequence formula to find specific terms. Here’s how to apply it:

Consider the step-by-step substitutions in the sequence formula:

• First, substitute \(n = 1\) into the formula: \(a_1 = 2(1) - 1 = 1\). So, the first term is 1.
• Next, substitute \(n = 2\): \(a_2 = 2(2) - 1 = 3\). So, the second term is 3.
• Continue with \(n = 3\): \(a_3 = 2(3) - 1 = 5\). The third term is 5.
• Finally, substitute \(n = 4\): \(a_4 = 2(4) - 1 = 7\). This results in the fourth term being 7.

Substitution simplifies the otherwise complex calculation by breaking it into manageable steps.

Term Calculation

Calculating terms involves specific steps, as shown in the example sequence. Here's a detailed breakdown:

Use the sequence formula \(a_n = 2n - 1\) and substitute different values of \(n\):

• For \(n = 1\): \(a_1 = 1\).
• For \(n = 2\): \(a_2 = 3\).
• For \(n = 3\): \(a_3 = 5\).
• For \(n = 4\): \(a_4 = 7\).

This process generates the sequence's terms: 1, 3, 5, 7.

Term calculation is straightforward when following these steps. Simply use the sequence formula for substitution, and list each calculated term. This systematic approach helps in visualizing the sequence as well.