Problem 2 - A Graphical Approach to Precalculus with Limits

Question

Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=6 n-3$$

Step-by-Step Solution

Answer

The first five terms are 3, 9, 15, 21, and 27.

Step 1: Understand the Sequence Formula

The sequence is given by the formula \(a_n = 6n - 3\), where \(n\) represents the term number. Our task is to find the first five terms by substituting \(n = 1, 2, 3, 4,\) and \(5\) into this formula.

Step 2: Calculate the First Term

Substitute \(n = 1\) into the sequence formula: \(a_1 = 6(1) - 3 = 6 - 3 = 3\). Thus, the first term is 3.

Step 3: Calculate the Second Term

Substitute \(n = 2\) into the sequence formula: \(a_2 = 6(2) - 3 = 12 - 3 = 9\). Thus, the second term is 9.

Step 4: Calculate the Third Term

Substitute \(n = 3\) into the sequence formula: \(a_3 = 6(3) - 3 = 18 - 3 = 15\). Thus, the third term is 15.

Step 5: Calculate the Fourth Term

Substitute \(n = 4\) into the sequence formula: \(a_4 = 6(4) - 3 = 24 - 3 = 21\). Thus, the fourth term is 21.

Step 6: Calculate the Fifth Term

Substitute \(n = 5\) into the sequence formula: \(a_5 = 6(5) - 3 = 30 - 3 = 27\). Thus, the fifth term is 27.

Key Concepts

Understanding the Sequence Formula

A sequence formula is a mathematical expression that describes how each term in a sequence is calculated based on its position. In arithmetic sequences, each term is produced by adding a constant difference to the previous term. For the formula given in this exercise, \(a_n = 6n - 3\), each term \(a_n\) is calculated based on its position \(n\).

• The term number \(n\) is crucial as it tells us where the term is located in the sequence.
• The sequence formula is a straightforward calculation where you can substitute different values of \(n\) and find the specific term at that position.

Understanding sequence formulas allows you to generate any term in the sequence systematically, without needing to list out all previous terms. It also simplifies your work when you need to find terms far into the sequence.

Steps in Term Calculation

Term calculation involves using the sequence formula to find specific terms. It's a simple process that requires basic arithmetic operations once you've substituted the term number into the formula.

• Start with the sequence formula: for this problem, we have \(a_n = 6n - 3\).
• To find each term, substitute \(n\) with the desired term number.

For example, to find the first term, substitute \(n = 1\):

• \(a_1 = 6(1) - 3 = 3\).

Calculate similarly for further terms:

• Second term: \(a_2 = 6(2) - 3 = 9\)
• Third term: \(a_3 = 6(3) - 3 = 15\)
• Fourth term: \(a_4 = 6(4) - 3 = 21\)
• Fifth term: \(a_5 = 6(5) - 3 = 27\)

By following these steps, you can easily calculate all required terms.

Using the Substitution Method in Arithmetic Sequences

The substitution method is a simple yet powerful algebraic technique often used in sequences and series. It helps calculate specific sequence terms without having to list preceding terms. This exercise's sequence formula, \(a_n = 6n - 3\), is perfect for substitution.

• Substitute the term number \(n\) into the sequence formula directly.
• Perform the arithmetic operations in the formula after substitution.

For instance, to find the fourth term \(a_4\), substitute \(n = 4\):

• Perform calculation: \(a_4 = 6 \cdot 4 - 3\).
• Solve: \(24 - 3 = 21\).

This method is efficient and reduces the chance of making errors as compared to manually listing terms. It is especially helpful for finding terms that are far into the sequence, without calculating each prior term.