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Chapter 14: Problem 88

Find a general term, \(a_{n},\) for each sequence. More than one answer may be possible. $$4,-8,16,-32, \dots$$

Short Answer

Expert verified
The general term for the given sequence is \(a_{n} = 4 \cdot (-2)^{n-1}\).

Step by step solution

01

Identifying the Pattern

Look at the provided sequence \(4,-8,16,-32, \dots\) and identify the pattern. The sequence starts at \(4\), and each subsequent term seems to be the previous term multiplied by \(-2\).
02

Formulating a General Rule

Our goal is to express this pattern in terms of \(n\), the term number. The first term (when \(n=1\)) is \(4\), and each subsequent term is \(-2\) times the previous term. We can formulate this as a rule for finding any term in the sequence: \(a_{n} = 4 \cdot (-2)^{n-1}\).
03

Testing the General Rule

We can validate our rule by using it to compute some of the sequence terms and check them against the given sequence. Given that our sequence starts with \(4,-8,16,-32, \dots\), let's try \(n=1, 2, 3, 4, 5\)... If the results match the given sequence, our rule is correct.
04

Concluding the Solution

Summary: We conducted an analysis of the sequence and identified that each term is \(-2\) times the previous term. We wrote a general term \(a_{n} = 4 \cdot (-2)^{n-1}\) to express this pattern. After a testing, we confirmed that the rule is correct. Therefore, the general term for the sequence is \(a_{n} = 4 \cdot (-2)^{n-1}\).

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Key Concepts

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

Geometric Sequences
A geometric sequence is a set of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." In this exercise, the sequence given is
  • 4, -8, 16, -32, ...
It shows a clear pattern where each term is multiplied by
  • -2
relative to the one before it.

This common ratio of
  • -2
allows us to easily deduce the sequence is geometric. Recognizing this is crucial because it allows us to quickly identify the general form of any term in the sequence.

The formula for the general term of a geometric sequence is given by the formula
  • \( a_{n} = a_{1} imes r^{(n-1)} \)
where
  • \( a_{1} \)
is the first term, and
  • \( r \)
is the common ratio. By substituting
  • \( a_{1} = 4 \) and \( r = -2 \)
we derive the formula for this particular sequence, which is
  • \( a_{n} = 4 imes (-2)^{n-1} \).
Pattern Recognition
Pattern recognition plays an important role in understanding geometric sequences. By simply observing the pattern, you can identify key characteristics such as the starting value and the recurring change, known as the common ratio.

In the sequence
  • 4, -8, 16, -32...
you can observe the signs of the terms alternate between positive and negative. This alternation is characteristic of sequences with a negative common ratio.

Recognizing such patterns allows us to visualize and predict the upcoming terms in the sequence intuitively. For example, knowing the term -32, you can instantly predict that the following number will be 64, simply by multiplying -32 by -2.

Grasping this concept is essential as it leads to constructing the general rule for the sequence. It involves focusing on both the numbers' values and how they change relative to one another.
Formula Derivation
Deriving the formula for a sequence involves transforming the recognized pattern into a mathematical equation that can predict any term. Following the recognition of a geometric sequence, like
  • 4, -8, 16, -32,...
you start by assuming a general form
  • \( a_{n} = a_{1} imes r^{(n-1)} \)


To derive the formula, determine:
  • The first term, \( a_{1} = 4 \)
  • The common ratio, \( r = -2 \)
Subsequently, plug these values into the formula to get the sequence's equation:
  • \[ a_{n} = 4 \times (-2)^{n-1} \]
This formula now provides a reliable way to calculate any term in the sequence by substituting the term number for
  • \( n \)


Formula derivation doesn't just stop at finding the rule; it's about understanding how the rule can consistently represent the sequence's behavior, enabling predictions of future terms accurately. This process encapsulates both pattern recognition and arithmetic operations in mathematics.

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Most popular questions from this chapter

Use the formula for the sum of the first n terms of a geometric sequence to solve. A job pays a salary of \(\$ 24,000\) the first year. During the next 19 years, the salary increases by \(5 \%\) each year. What is the total lifetime salary over the 20 -year period? Round to the nearest dollar.

Each exercise involves observing a pattern in the expanded form of the binomial expression \((a+b)^{n}\).$$\begin{array}{l}(a+b)^{1}=a+b \\\\(a+b)^{2}=a^{2}+2 a b+b^{2} \\\\(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3} \\\\(a+b)^{4}=a^{4}+4 a^{3} b+6 a^{2} b^{2}+4 a b^{3}+b^{4} \\\\(a+b)^{5}=a^{5}+5 a^{4} b+10 a^{3} b^{2}+10 a^{2} b^{3}+5 a b^{4}+b^{5}\end{array}$$ Describe the pattern for the sum of the exponents on the variables in each term.

Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{2 n^{2}+5 n-7}{n^{3}}, n:[0,10,1] \text { by } a_{n}:[0,2,0.2]$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\sum_{i=1}^{4} 3 i+\sum_{i=1}^{4} 4 i=\sum_{i=1}^{4} 7 i$$

Use the formula for the general term (the nth term) of a geometric sequence to solve. A professional baseball player signs a contract with a beginning salary of \(\$ 3,000,000\) for the first year and an annual increase of \(4 \%\) per year beginning in the second year. That is, beginning in year \(2,\) the athlete's salary will be 1.04 times what it was in the previous year. What is the athlete's salary for year 7 of the contract? Round to the nearest dollar.
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