Problem 3 - Precalculus

Question

Write out the first four terms of the given sequence. \(\\{5 k-2\\}_{k=1}^{\infty}\)

Step-by-Step Solution

Answer

The first four terms are 3, 8, 13, and 18.

Step 1: Understand the Sequence Rule

The sequence is given by \(\{5k - 2\}_{k=1}^{\infty}\). This means for each position \(k\), the term is calculated using the formula \(5k - 2\). We need the first four terms where \(k=1, 2, 3, 4\).

Step 2: Calculate the First Term

Substitute \(k=1\) into the sequence formula: \(5(1) - 2 = 5 - 2 = 3\). The first term is \(3\).

Step 3: Calculate the Second Term

Substitute \(k=2\) into the sequence formula: \(5(2) - 2 = 10 - 2 = 8\). The second term is \(8\).

Step 4: Calculate the Third Term

Substitute \(k=3\) into the sequence formula: \(5(3) - 2 = 15 - 2 = 13\). The third term is \(13\).

Step 5: Calculate the Fourth Term

Substitute \(k=4\) into the sequence formula: \(5(4) - 2 = 20 - 2 = 18\). The fourth term is \(18\).

Key Concepts

Sequence Terms

In mathematics, a **sequence** is an ordered list of numbers. Each number in this list is called a **term**. Understanding sequence terms is crucial, because they help you identify patterns and understand how numbers are organized within a sequence.

In the sequence given by the formula \(5k - 2\), for instance, each term is calculated using a specific value for \(k\), which represents the position in the sequence.

Here is how we determine the sequence terms from the formula:

• When \(k = 1\), the result is the first term.
• When \(k = 2\), the result is the second term.
• When \(k = 3\), the result is the third term.
• When \(k = 4\), the result is the fourth term.

Each term becomes part of a larger pattern that follows the rule \(5k - 2\). This structured approach gives us insight into how sequences behave.

Arithmetic Sequence

An **arithmetic sequence** is a type of sequence where each term after the first is found by adding a constant to the previous term.

In the sequence \(5k - 2\), each term increases by a fixed number as \(k\) increases by 1. To identify this constant difference, you can calculate the difference between consecutive terms.

• For example, starting with the first term, 3: \(8 - 3 = 5\) is the difference between the second and first term.
• Again, \(13 - 8 = 5\), the difference between the third and second term.
• Finally, \(18 - 13 = 5\), the difference between the fourth and third term.

This constant difference of 5 shows that the sequence is indeed arithmetic. Understanding arithmetic sequences helps students recognize the linear pattern and predict future terms.

Algebraic Expressions

An **algebraic expression** is a mathematical phrase that includes numbers, variables, and operations. The expression \(5k - 2\) is a good example. Here, "5" is a coefficient, "k" is the variable, and "-2" is a constant.

This expression represents a rule or formula for the sequence, indicating how each term is derived. To work with algebraic expressions effectively, it's important to:

• Understand what each part of the expression represents.
• Substitute values for variables (like \(k\)) to compute specific terms.
• Recognize how changes in the variable affect the outcome.

Through practice, you can become comfortable manipulating algebraic expressions. They represent not only individual sequence terms but broader mathematical relationships as well.