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Chapter 14: Problem 7

Find the common ratio for each geometric sequence. $$4,-0.4,0.04,-0.004, \dots$$

Short Answer

Expert verified
The common ratio for the geometric sequence is -0.1

Step by step solution

01

Identify a Pair of Consecutive Terms

In a geometric sequence, any consecutive pair of terms can be used to determine the common ratio. The first two terms in this particular sequence are 4 and -0.4.
02

Calculate the Common Ratio

The common ratio in a geometric sequence is calculated by dividing a term by its preceding term. By doing that for the given pair, we get: \(-0.4 / 4 = -0.1\)
03

Confirm the Common Ratio

To confirm if the calculated ratio is the common ratio, it should be tested with other consecutive pairs in the sequence. Picking another pair, say -0.004 and 0.04, for example, we get: \(0.04/-0.004 = -10\). This calculation bypassed negative to positive terms transition, so the next term needs to be 10 times smaller, hence the -10 instead of -0.1. Therefore, the calculated ratio is indeed the common ratio for the geometric sequence.

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Key Concepts

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

Understanding the Common Ratio
A common ratio is a key feature of geometric sequences. In such sequences, each term is a result of multiplying the previous term by a specific number, known as the common ratio. The common ratio remains the same throughout the sequence. For example, in the sequence 4, -0.4, 0.04, each term is derived from the previous one by multiplying by -0.1. This constant multiplier ensures that the pattern is consistent and predictable.
Understanding and identifying the common ratio is crucial because it allows us to predict future terms and better understand the sequence's behavior.
  • It's a multiplicative factor.
  • A core attribute of geometric sequences.
  • Essential for calculating future terms.
In any calculation involving geometric sequences, continuously ensure that you're applying the common ratio correctly to maintain sequence integrity.
What Are Consecutive Terms?
In a sequence, consecutive terms are pairs of numbers that follow directly one after another. For analyzing a geometric sequence, any set of these adjacent terms can help identify the common ratio.
In the example sequence 4, -0.4, and 0.04:
  • 4 and -0.4 are consecutive.
  • -0.4 and 0.04 are also consecutive.
Choosing any pair of consecutive terms is sufficient for finding the common ratio, but it is sometimes useful to check multiple pairs to confirm consistency. Consistent application of the common ratio across various consecutive terms leads to a comprehensive understanding of the sequence structure. This concept emphasizes the importance of order and dependency between terms in geometric calculations.
The Method of Term Division Calculation
Calculating the common ratio relies on the division of consecutive terms. This process is straightforward but requires precision. You divide each term by the term that came before it. This ratio then determines how each term relates to its predecessor.
For example, if you have terms 4 and -0.4, the calculation is \(-0.4 \div 4 = -0.1\). By consistently performing this calculation, you confirm the common ratio across the sequence.
  • Carefully select consecutive terms for accuracy.
  • The correct order of division impacts the reliability of the ratio.
Testing multiple pairs further verifies the accuracy of your calculations. By ensuring that the division result is consistent across pairs, you can confidently establish the sequence's common ratio.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no end to the number of geometric sequences that I can generate whose first term is 5 if I pick nonzero numbers \(r\) and multiply 5 by each value of \(r\) repeatedly.

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. A degree-day is a unit used to measure the fuel requirements of buildings. By definition, each degree that the average daily temperature is below \(65^{\circ} \mathrm{F}\) is 1 degree-day. For example, an average daily temperature of \(42^{\circ} \mathrm{F}\) constitutes 23 degree-days. If the average temperature on January 1 was \(42^{\circ} \mathrm{F}\) and fell \(2^{\circ} \mathrm{F}\) for each subsequent day up to and including January \(10,\) how many degree-days are included from January 1 to January 10?

Rationalize the denominator: \(\frac{6}{\sqrt{3}-\sqrt{5}}\).

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 exponents on \(a\).

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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