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Chapter 5: Problem 108

Determine whether each sequence is arithmetic or geometric. Then find the next two terms. \(\frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}, \ldots\)

Short Answer

Expert verified
The given sequence is an arithmetic sequence. The next two terms in the sequence are \(2\) and \(\frac{7}{3}\).

Step by step solution

01

Identify if the Sequence is Arithmetic or Geometric

To determine whether the sequence is arithmetic or geometric, examine the difference and ratio between consecutive terms. For the given sequence: \(\frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}\), the differences are: \(1 - \frac{2}{3} = \frac{1}{3}\), \(\frac{4}{3} - 1 = \frac{1}{3}\), \(\frac{5}{3} - \frac{4}{3} = \frac{1}{3}\). So, the sequence has a common difference, which means it is an arithmetic sequence.
02

Find the Next Two Terms

As the given sequence is an arithmetic sequence and we have identified the common difference to be \(\frac{1}{3}\), the next terms can be found by adding the common difference to the previous terms. Therefore, the next term after \(\frac{5}{3}\) would be \(\frac{5}{3} + \frac{1}{3} = 2\) and the term after that would be \(2 + \frac{1}{3} = \frac{7}{3}\). Thus, the next two terms in the sequence would be \(2\) and \(\frac{7}{3}\).

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Key Concepts

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

Arithmetic Sequence
An arithmetic sequence is a list of numbers where the difference between consecutive terms is always the same. This consistent difference is what defines the sequence as 'arithmetic'.

If you look at the example sequence \(\frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}\), you can easily spot how each term is obtained. You simply add a fixed number to the previous term to get the next one.

For this particular sequence, if you calculate \(1 - \frac{2}{3}\), \(\frac{4}{3} - 1\), and \(\frac{5}{3} - \frac{4}{3}\), they all equal \(\frac{1}{3}\). This equal difference confirms we have an arithmetic sequence. Every term follows a neat, predictable pattern.
Common Difference
The common difference in an arithmetic sequence is the value you add to each term to move to the next one. It is what makes the sequence linear and predictable. Knowing the common difference allows you to extend the sequence forwards or backwards with ease.

To find this value, as shown in our example, you subtract a term from the one following it. In the sequence \(\frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}\), each successive subtraction gives us \(\frac{1}{3}\).

Understanding the common difference encloses the logic of an arithmetic sequence. It is crucial for determining future terms without recalculating each interval. Simply continue adding the common difference to the latest known term.
Geometric Sequence
A geometric sequence, often confused with arithmetic, involves multiplying (or dividing) by a fixed number called the common ratio, rather than adding or subtracting a fixed amount. This makes for a different kind of pattern, one that grows or shrinks exponentially.

Unlike our exercise's arithmetic set, where addition forms the sequence, a geometric sequence results from something like \(2, 6, 18, 54\), where each term is multiplied by 3 to get the next. Here, 3 is the common ratio.

Geometric sequences open up a world of exponentially growing patterns, contrasting the linear growth we're presented with in our arithmetic example. Remember, identifying whether a sequence is arithmetic or geometric involves checking if the changes between terms are about addition or multiplication.

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Most popular questions from this chapter

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=-4, r=-2\)

The sum, \(S_{n}\), of the first \(n\) terms of an arithmetic sequence is given by$$S_{n}=\frac{n}{2}\left(a_{1}+a_{n}\right),$$in which \(a_{1}\) is the first term and \(a_{n}\) is the nth term. The sum, \(S_{n}\), of the first \(n\) terms of a geometric sequence is given by$$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r},$$in which \(a_{1}\) is the first term and \(r\) is the common ratio \((r \neq 1)\). Determine whether each sequence is arithmetic or geometric. Then use the appropriate formula to find \(S_{10}\), the sum of the first ten terms. \(3,-6,12,-24, \ldots\)

What is the common difference in an arithmetic sequence?

Write the first six terms of the geometric sequence with the first term, \(a_{1}\), and common ratio, \(r\). \(a_{1}=2, r=3\)

You will develop geometric sequences that model the population growth for California and Texas, the two most populated U.S. states. The table shows the population of Texas for 2000 and 2010 , with estimates given by the U.S. Census Bureau for 2001 through \(2009 .\) $$ \begin{aligned} &\begin{array}{|l|c|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 0} & \mathbf{2 0 0 1} & \mathbf{2 0 0 2} & \mathbf{2 0 0 3} & \mathbf{2 0 0 4} & \mathbf{2 0 0 5} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 20.85 & 21.27 & 21.70 & 22.13 & 22.57 & 23.02 \\ \hline \end{array}\\\ &\begin{array}{|l|c|c|c|c|c|} \hline \text { Year } & \mathbf{2 0 0 6} & \mathbf{2 0 0 7} & \mathbf{2 0 0 8} & \mathbf{2 0 0 9} & \mathbf{2 0 1 0} \\ \hline \begin{array}{l} \text { Population } \\ \text { in millions } \end{array} & 23.48 & 23.95 & 24.43 & 24.92 & 25.15 \\ \hline \end{array} \end{aligned} $$ a. Divide the population for each year by the population in the preceding year. Round to two decimal places and show that Texas has a population increase that is approximately geometric. b. Write the general term of the geometric sequence modeling Texas's population, in millions, \(n\) years after \(1999 .\) c. Use your model from part (b) to project Texas's population, in millions, for the year 2020 . Round to two decimal places.
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