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Chapter 9: Problem 3

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

Short Answer

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

Step by step solution

01

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 \(k=1, 2, 3, 4\).
02

Calculate the First Term

Substitute \(k=1\) into the sequence formula: \(5(1) - 2 = 5 - 2 = 3\). The first term is \(3\).
03

Calculate the Second Term

Substitute \(k=2\) into the sequence formula: \(5(2) - 2 = 10 - 2 = 8\). The second term is \(8\).
04

Calculate the Third Term

Substitute \(k=3\) into the sequence formula: \(5(3) - 2 = 15 - 2 = 13\). The third term is \(13\).
05

Calculate the Fourth Term

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

In Exercises \(18-22,\) use the Binomial Theorem to find the indicated term. The term containing \(x^{\frac{7}{2}}\) in the expansion \((\sqrt{x}-3)^{8}\)

Prove each assertion using the Principle of Mathematical Induction. \(2^{n}>500 n\) for \(n>12\)

Find an explicit formula for the \(n^{\text {th }}\) term of the given sequence. Use the formulas in Equation 9.1 as needed. \(27,64,125,216, \ldots\)

In Exercises \(1-9\), simplify the given expression. $$ \left(\begin{array}{l} 8 \\ 3 \end{array}\right) $$

You've just won three tickets to see the new film, \(8.5 .\) Five of your friends, Albert, Beth, Chuck, Dan, and Eugene, are interested in seeing it with you. With the help of your classmates, list all the possible ways to distribute your two extra tickets among your five friends. Now suppose you've come down with the flu. List all the different ways you can distribute the three tickets among these five friends. How does this compare with the first list you made? What does this have to do with the fact that \(\left(\begin{array}{l}5 \\ 2\end{array}\right)=\left(\begin{array}{l}5 \\\ 3\end{array}\right)\) ?
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