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Chapter 9: Problem 22

In Exercises 17 - 28, write the first five terms of the geometric sequence \( a_1 = 6, r = -\dfrac{1}{4} \)

Short Answer

Expert verified
The first five terms of the geometric sequence are: 6, -3/2, 3/2, -3/8, 3/16.

Step by step solution

01

Calculate the first term

The first term is given, which is \(a_1 = 6\).
02

Calculate the second term

Use the formula with \(i = 2\). So the second term is \(a_2 = a_1 \cdot r^{2-1} = 6 \cdot -\dfrac{1}{4} = -\dfrac{3}{2}\).
03

Calculate the third term

Use the formula with \(i = 3\). So the third term is \(a_3 = a_1 \cdot r^{3-1} = 6 \cdot \left(-\dfrac{1}{4}\right)^2 = \dfrac{3}{2}\).
04

Calculate the fourth term

Use the formula with \(i = 4\). So the fourth term is \(a_4 = a_1 \cdot r^{4-1} = 6 \cdot \left(-\dfrac{1}{4}\right)^3 = -\dfrac{3}{8}\).
05

Calculate the fifth term

Use the formula with \(i = 5\). So the fifth term is \(a_5 = a_1 \cdot r^{5-1} = 6 \cdot \left(-\dfrac{1}{4}\right)^4 = \dfrac{3}{16}\).

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Key Concepts

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

first five terms
To understand a geometric sequence, you need to know how to determine its first few terms. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

In our example, we begin with the first term, which is given as 6. Then, we use the common ratio of \( -\frac{1}{4} \) to find the following terms.
  • The second term: Multiply the first term by the common ratio, \( 6 \times -\frac{1}{4} = -\frac{3}{2} \).
  • The third term: Continue by multiplying the second term by the common ratio, \( -\frac{3}{2} \times -\frac{1}{4} = \frac{3}{2} \).
  • The fourth term: Again, multiply the previous term by the common ratio, \( \frac{3}{2} \times -\frac{1}{4} = -\frac{3}{8} \).
  • The fifth term: Follow the pattern for one more term, \( -\frac{3}{8} \times -\frac{1}{4} = \frac{3}{16} \).
By following this pattern, you can easily find the first five terms of a geometric sequence.
common ratio
The common ratio is the backbone of a geometric sequence. It's the constant factor that each term is multiplied by to produce the next term in the series. In our example, the common ratio is given as \(-\frac{1}{4}\).

Understanding the role of the common ratio is crucial:
  • A positive common ratio means the sequence terms will always have the same sign if the first term is positive.
  • A negative common ratio, like our example, causes the signs of the terms to alternate between positive and negative.
  • If the common ratio is greater than 1, the sequence will grow larger with each term.
  • If the common ratio is between -1 and 1, like \(-\frac{1}{4}\), the terms will decrease in absolute value.
With a firm grasp of the common ratio, you can predict how the sequence behaves and generates new terms.
sequence formula
A special formula helps in finding any term of a geometric sequence without listing all previous terms. This is particularly helpful for sequences with many terms.

For a geometric sequence, the n-th term \(a_n\) can be calculated using the formula:\[a_n = a_1 \, \cdot \, r^{n-1}\]Where:
  • \(a_1\) is the first term.
  • \(r\) is the common ratio.
  • \(n\) is the term number.
Using our original exercise, to find the second term, we set \(n = 2\):\[a_2 = 6 \, \cdot \, \left(-\frac{1}{4}\right)^{2-1} = -\frac{3}{2}\]This formula saves time and energy, allowing you to quickly calculate any term in the sequence directly. It's a powerful tool for mastering geometric sequences.

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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 three consecutive spins.

In Exercises 53 - 60, the sample spaces are large and you should use the counting principles discussed in Section 9.6. A shipment of \( 12 \) microwave ovens contains three defective units. A vending company has ordered four of these units, and because each is identically packaged, the selection will be random. What are the probabilities that (a) all four units are good,(b) exactly two units are good, and (c) at least two units are good?

Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in \( 30 \) states, Washington D.C., and the U.S. Virgin Islands.The game is played by drawing five white balls out of a drum of \( 59 \) white balls (numbered \( 1 - 59 \)) and one red powerball out of a drum of \( 39 \) red balls (numbered \( 1 - 39 \)). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers if the jackpot is won by matching all five white balls in order and the red power ball. (c) Compare the results of part (a) with a state lottery in which a jackpot is won by matching six balls from a drum of \( 59 \) balls.

In Exercises 15 - 20, find the probability for the experiment of tossing a coin three times. Use the sample space \( S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} \). The probability of getting exactly two tails

In Exercises 25 - 30, find the probability for the experiment of tossing a six-sided die twice. The sum is less than \( 11 \).
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