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Chapter 9: Problem 9

Find the \(n\) th term, the fifth term, and the eighth term of the geometric sequence. $$4,-6,9,-13.5, \dots$$

Short Answer

Expert verified
General formula: \( a_n = 4(-1.5)^{n-1} \); fifth term: 20.25; eighth term: -68.34375.

Step by step solution

01

Understand Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, denoted by \( r \). In this sequence, the terms are 4, -6, 9, -13.5.
02

Calculate the Common Ratio

To find the common ratio \( r \), divide the second term by the first term. \[ r = \frac{-6}{4} = -1.5 \] Verify by checking other term ratios: \( \frac{9}{-6} = -1.5 \) and \( \frac{-13.5}{9} = -1.5 \). The common ratio \( r = -1.5 \) holds true.
03

Write the General Term Formula

For a geometric sequence, the \( n \)th term \( a_n \) is given by \[ a_n = a_1 \times r^{(n-1)} \] where \( a_1 \) is the first term and \( r \) is the common ratio. Here, \( a_1 = 4 \) and \( r = -1.5 \).
04

Find the n-th Term Formula

Plug the values into the general formula: \[ a_n = 4 \times (-1.5)^{(n-1)} \]. This is the formula for the \( n \)th term of the sequence.
05

Calculate the Fifth Term

Use the formula for the fifth term where \( n = 5 \): \[ a_5 = 4 \times (-1.5)^{4} \]. Calculate \((-1.5)^4 = 5.0625\), so, \[ a_5 = 4 \times 5.0625 = 20.25 \].
06

Calculate the Eighth Term

Use the formula for the eighth term where \( n = 8 \): \[ a_8 = 4 \times (-1.5)^{7} \]. Calculate \((-1.5)^7 = -17.0859375\), so, \[ a_8 = 4 \times -17.0859375 = -68.34375 \].

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Key Concepts

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

Common Ratio
In any geometric sequence, the common ratio is the constant factor between two consecutive terms. This ratio is denoted by \( r \) and is critical in determining the behavior of the sequence. To find the common ratio, take any two consecutive terms in the sequence and divide the second term by the first.
For example, with the sequence 4, -6, 9, -13.5, the common ratio \( r \) can be calculated as follows:
  • Take the second term \(-6\) and divide it by the first term \(4\):
  • \( r = \frac{-6}{4} = -1.5 \)
Verification with subsequent terms, such as \( \frac{9}{-6} \) and \( \frac{-13.5}{9} \), shows that the common ratio \( r = -1.5 \) is consistent. This means that every term in this sequence is 1.5 times the preceding term, with a change in direction due to the negative sign.
n-th Term
The \( n \)-th term formula of a geometric sequence is used to find any term in the sequence without having to list all the previous terms. The formula for the \( n \)-th term \( a_n \) is expressed as:
  • \( a_n = a_1 \times r^{(n-1)} \)
where \( a_1 \) is the first term, and \( r \) is the common ratio.
In our sequence, the formula becomes \( a_n = 4 \times (-1.5)^{n-1} \). This equation allows us to find any term position in the sequence by plugging in the value of \( n \). It captures the exponential growth or decay pattern, characterized by the powers of the common ratio.
Fifth Term
The fifth term of a geometric sequence corresponds to the position \( n = 5 \). Using our \( n \)-th term formula, \( a_5 = 4 \times (-1.5)^{4} \). Let's break it down:
  • The expression \((-1.5)^4\) indicates raising the common ratio to the power of four, as you move four steps ahead of the first term.
  • \((-1.5)^4 = 5.0625\).
  • Then multiply by the first term to find \( a_5 = 4 \times 5.0625 = 20.25 \).
By following these steps, you can determine that the fifth term in this sequence is 20.25, emphasizing the multiplicative pattern over the arithmetic succession.
Eighth Term
Finding the eighth term involves applying the \( n \)-th term formula at \( n = 8 \). This tells us:
  • \( a_8 = 4 \times (-1.5)^{7} \).
Here's a closer look:
  • Calculate \((-1.5)^7\), which equals \(-17.0859375\). This is essential as it shows the continued influence of negative multiplication over the sequence.
  • Then, multiplying by the first term, we have \( a_8 = 4 \times -17.0859375 = -68.34375 \).
Thus, the eighth term is \(-68.34375\), demonstrating how geometric sequences can dramatically increase or decrease depending on the sign and value of the common ratio.

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Most popular questions from this chapter

A company is to distribute \(\$ 46,000\) in bonuses to its top ten salespeople. The tenth salesperson on the list will receive \(\$ 1000,\) and the difference in bonus money between successively ranked salespeople is to be constant. Find the bonus for each salesperson.

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Find the number of terms in the arithmetic sequence with the given conditions. $$a_{1}=-1, \quad d=\frac{1}{5}, \quad S=21$$

Find the sum. $$\sum_{k=1}^{6} \frac{3}{k+1}$$

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