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Chapter 9: Problem 23

Writing the \(n\) th Term of a Geometric Sequence, write the first five terms of the geometric sequence. Determine the common ratio and write the \(n\) th term of the sequence as a function of \(n .\) $$ a_{1}=64, \quad a_{k+1}=\frac{1}{2} a_{k} $$

Short Answer

Expert verified
The first five terms of the sequence are 64, 32, 16, 8, 4. The common ratio of the sequence is \(0.5 = 1/2\). The \(n\)th term of the sequence is given by the formula \(a_n = 64 \cdot (0.5)^{n-1}\).

Step by step solution

01

Calculation of the First Five Terms

Use the recursive formula \( a_{k+1} \frac{1}{2} a_k \) and the fact that \( a_1 = 64 \) to find the first five terms of the geometric sequence. \[ a_1 = 64 \] \[ a_2 = 64/2 = 32 \] \[ a_3 = 32/2 = 16 \] \[ a_4 = 16/2 = 8 \] \[ a_5 = 8/2 = 4 \]
02

Identifying the Common Ratio

By simple observation, we can see that each term is obtained by dividing the previous term by 2. Thus, the common ratio \( r \) is \( 1/2 = 0.5 \).
03

Writing the nth Term of the Sequence

The general form of a geometric sequence is \( a_n = a_1 \cdot r^{(n-1)} \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Substituting the values we have: \( a_1 = 64 \) and \( r = 0.5 \), we get the \( n \) th term as: \( a_n = 64 \cdot 0.5^{(n-1)} \).

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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, one of the key features is the common ratio, denoted as r. The common ratio can be defined as the constant factor between consecutive terms in a geometric sequence. If we take two successive terms ak and ak+1, then the common ratio is obtained by dividing the latter by the former:

\[ r = \frac{a_{k+1}}{a_{k}} \]
In our exercise, with the first term being 64 and each subsequent term obtained by halving the previous, we can clearly see that the common ratio is \( \frac{1}{2} \). This means that every term in the sequence is half the value of the term that precedes it. Recognizing the common ratio is critical as it allows us to describe the behavior of the sequence over time, qualifying it as a model of exponential decay in this case.
Recursive Formula
The recursive formula is an expression that defines each term of a sequence using one or more preceding terms. For a geometric sequence, the recursive formula typically takes the form:
\[ a_{k+1} = r \cdot a_{k} \],
where \( a_{k} \) is an existing term and \( a_{k+1} \) is the next term in the sequence. In our exercise example, the recursive formula was given as \( a_{k+1} = \frac{1}{2} a_{k} \),
indicating that each new term is half of its predecessor. This type of formula is particularly useful for understanding the step-by-step progression of a sequence and for calculating individual terms when the sequence is followed in order.
Nth Term of a Sequence
Determining the nth term of a sequence allows us to calculate any term in the sequence without the need to progress sequentially from one term to the next. For a geometric sequence, the nth term is given by the formula:
\[ a_n = a_1 \cdot r^{(n-1)} \]
With this formula, we can directly find the value of the nth term by knowing the first term \( a_1 \), the common ratio \( r \), and the position \( n \) we are interested in. This formula encapsulates the whole sequence into a single neat expression, making it a powerful tool for both understanding the sequence's overarching pattern and for practical calculations. In our specific exercise, substituting the values into the nth term formula, \( a_1 = 64 \) and \( r = 0.5 \), provides a direct way to calculate the value of any term in the sequence: \( a_n = 64 \cdot 0.5^{(n-1)} \).

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Most popular questions from this chapter

American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered \(1-36,\) of which half are red and half are black. Two of the pockets are green and are numbered 0 and 00 (see figure). The dealer spins the wheel and a small ball in opposite directions. As the ball slows to a stop, it has an equal probability of landing in any of the numbered pockets. (a) Find the probability of landing in the number 00 pocket. (b) Find the probability of landing in a red pocket. (c) Find the probability of landing in a green pocket or a black pocket. (d) Find the probability of landing in the number 14 pocket on two consecutive spins. (e) Find the probability of landing in a red pocket on \(\quad\) three consecutive spins.

Finding a Linear or Quadratic Model In Exercises \(55-60\) , decide whether the sequence can be represented perfectly by a linear or a quadratic model. If so, then find the model. $$0,6,16,30,48,70, \dots$$

Child Support The amounts \(f(t)\) (in billions of dollars) of child support collected in the United States from 2002 through 2009 can be approximated by the model $$f(t)=-0.009 t^{2}+1.05 t+18.0, \quad 2 \leq t \leq 9$$ where \(t\) represents the year, with \(t=2\) corresponding to \(2002 .\) (Source: U.S. Department of Health and Human Services) (a) You want to adjust the model so that \(t=2\) corresponds to 2007 rather than \(2002 .\) To do this you shift the graph of five units to the left to obtain \(g(t)=f(t+5) .\) Use binomial coefficients to write \(g(t)\) in standard form. (b) Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. (c) Use the graphs to estimate when the child support collections exceeded \(\$ 25\) billion.

Linear Model, Quadratic Model, or Neither? In Exercises \(61-68\) , write the first six terms of the sequence beginning with the given term. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither. $$a_{1}=0$$ $$a_{n}=a_{n-1}+2 n$$

Graphical Reasoning In Exercises 83 and \(84,\) use a graphing utility to graph \(f\) and \(g\) in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function \(g\) in standard form. $$f(x)=x^{3}-4 x, \quad g(x)=f(x+4)$$
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