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Chapter 10: Problem 44

Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) $$ 1+\frac{1}{2}, 1+\frac{1}{4}, 1+\frac{1}{8}, 1+\frac{1}{16}, \ldots $$

Short Answer

Expert verified
The \(n\) th term of the given sequence can be written as \(1 + \frac{1}{2^{(n-1)}}\).

Step by step solution

01

Identify Sequence Type

The first step is to identify the type of sequence. Here, the terms are getting divided by 2 each time, which indicates that this is a geometric sequence.
02

Identify Common Ratio

Next, find the common ratio of the geometric sequence. The ratio is obtained by dividing any term by the preceding term. In this case, it's \(\frac{1}{2}\), then \(\frac{1}{4}\), then \(\frac{1}{8}\), and so on. Each term is half of the previous term, which suggests the ratio is \(\frac{1}{2}\).
03

Write General Term

The general term of a geometric sequence is usually written as \(a*r^{(n-1)}\), where \(a\) stands for the first term, \(r\) is the common ratio, and \(n\) is the term number. Here the general term would look like: \(1 + \frac{1}{2^{(n-1)}}\). It means the \(n\) th term is obtained by adding 1 to the fraction whose denominator is an \(n-1\) th power of 2.

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Key Concepts

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

Common Ratio
In a geometric sequence, the common ratio is the factor by which you multiply one term to get the next term. For this sequence, each term is half of the previous one.
This means the common ratio is \( \frac{1}{2} \). To find the common ratio, divide any term by the term directly before it.
  • For example, dividing \(1+\frac{1}{4}\) by \(1+\frac{1}{2}\) simplifies to \(\frac{\frac{3}{4}}{\frac{3}{2}} = \frac{1}{2}\).
  • The common ratio works the same way across all pairs of successive terms in the sequence.
This ratio is crucial for understanding how the sequence progresses, forming the backbone of all further calculations.
General Term of Sequence
The general term of a sequence gives a formula to find any term in the sequence without listing all preceding terms. For our sequence, the first term is \(1 + \frac{1}{2}\) and the common ratio is \(\frac{1}{2}\).
To construct the general formula, use:
  • First term \(a = 1 + \frac{1}{2}\).
  • Common ratio \(r = \frac{1}{2}\).
The general form for this sequence is:
  • \(a_n = a \, r^{(n-1)}\)
Substituting the known values:
  • \(a_n = \left(1 + \frac{1}{2}\right) \left(\frac{1}{2}\right)^{(n-1)}\)
  • Or simplified to \(a_n = 1 + \frac{1}{2^{(n-1)}}\)
This formula allows you to find the value of any term, \(a_n\), in the sequence.
Nth Term Formula
The nth term formula helps in pinpointing the nth term directly. This can be particularly useful in large sequences, where listing terms isn't practical. The nth term formula for a geometric sequence is:
  • \(a_n = a_1 \, r^{(n-1)}\)
Here:
  • \(a_1 = 1 + \frac{1}{2}\)
  • \(r = \frac{1}{2}\)
  • The nth term formula results in: \(a_n = 1 + \frac{1}{2^{(n-1)}}\)
This formula efficiently jumps straight to any term you need, enhancing understanding of sequence growth.
Notice that the denominator of the fraction increases exponentially by 2, showing the power and flexibility of using this formula in calculations.

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Most popular questions from this chapter

The repeating decimal is expressed as a geometric series. Find the sum of the geometric series and write the decimal as the ratio of two integers. $$ 0 . \overline{4}=0.4+0.04+0.004+0.0004+\cdots $$

Determine the convergence or divergence of the series. Use a symbolic algebra utility to verify your result. $$ \sum_{n=0}^{\infty} \frac{1}{4^{n}} $$

Carbon Dioxide The average concentration levels of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) in Earth's atmosphere for selectedyears since \(1980,\) in parts per million of carbon dioxide, are shown in the table. \(\quad(\text {Source: } N O A A)\). $$ \begin{array}{|c|c|c|c|c|c|c|}\hline n & {0} & {5} & {10} & {15} & {20} & {25} \\\ \hline a_{n} & {338.7} & {345.3} & {353.8} & {359.9} & {368.8} & {378.8} \\\ \hline\end{array} $$ (a) Use the regression feature of a graphing utility to find a model of the form \(a_{n}=k n+b\) for the data. Let \(n\) represent the year, with \(n=0\) corresponding to \(1980 .\) Use a graphing utility to plot the points and graph the model. (b) Use the model to predict the average concentration level of \(\mathrm{CO}_{2}\) in the year 2015 .

Write the next two terms of the arithmetic sequence. Describe the pattern you used to find these terms. $$ \frac{1}{2}, \frac{5}{4}, 2, \frac{11}{4}, \ldots $$

Depreciation A company buys a machine for 225,000 dollars that depreciates at a rate of \(30 \%\) per year. Find a formula for the value of the machine after \(n\) years. What is its value after 5 years?
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