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Magazine Chapter 9: Problem 10
Find the \(n\) th term, the fifth term, and the elghth term of the geometric sequence. $$162,-54,18,-6, \dots$$
Short Answer
Expert verified
The nth term is \(162(-\frac{1}{3})^{n-1}\), 5th term is 2, 8th term is \(-\frac{2}{27}\).
Step by step solution
01
Identify the First Term and Common Ratio
In a geometric sequence, each term after the first is the product of the previous term and the common ratio. The first term, denoted as \(a\), is 162. To find the common ratio \(r\), divide the second term by the first term: \( r = \frac{-54}{162} = -\frac{1}{3} \).
02
Find the General Formula for the nth Term
The general formula for the \(n\)th term \(a_n\) of a geometric sequence is: \[ a_n = a \cdot r^{n-1} \]Substitute \(a = 162\) and \(r = -\frac{1}{3}\): \[ a_n = 162 \left(-\frac{1}{3}\right)^{n-1} \]
03
Calculate the Fifth Term
Substitute \(n = 5\) into the formula to find the fifth term: \[ a_5 = 162 \left(-\frac{1}{3}\right)^{5-1} = 162 \left(-\frac{1}{3}\right)^4 = 162 \times \frac{1}{81} = 2 \]
04
Calculate the Eighth Term
Substitute \(n = 8\) into the formula to find the eighth term: \[ a_8 = 162 \left(-\frac{1}{3}\right)^{8-1} = 162 \left(-\frac{1}{3}\right)^7 = 162 \times \left(-\frac{1}{2187}\right) = -\frac{54}{729} = -\frac{2}{27} \]
05
Summary of Calculated Terms
The formula for the \(n\)th term is \(162 \left(-\frac{1}{3}\right)^{n-1}\). The fifth term is \(2\) and the eighth term is \(-\frac{2}{27}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Ratio
A geometric sequence is a series of numbers where each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the "common ratio." Understanding how to find this common ratio is a key step in working with geometric sequences.
This ratio can be easily determined by dividing any term in the sequence by its preceding term.
For the sequence given in the exercise, the first term is 162, and the second term is -54. Thus, the common ratio \( r \) can be calculated as follows:
This ratio can be easily determined by dividing any term in the sequence by its preceding term.
For the sequence given in the exercise, the first term is 162, and the second term is -54. Thus, the common ratio \( r \) can be calculated as follows:
- Choose any two consecutive terms, such as -54 and 162
- Divide the second term by the first term: \( r = \frac{-54}{162} = -\frac{1}{3} \)
NTH Term Formula
The \(n\)th term formula is essential for finding any term in a geometric sequence without calculating all the previous terms. This formula provides a direct way to find, for example, the 50th or the 100th term of the sequence, with ease.
The general formulation for the \(n\)th term \(a_n\) in a geometric sequence is:
The general formulation for the \(n\)th term \(a_n\) in a geometric sequence is:
- \(a_n = a \cdot r^{n-1}\)
- \(a\) is the first term of the sequence
- \(r\) is the common ratio
- \(n\) is the term number
- \[ a_n = 162 \left(-\frac{1}{3}\right)^{n-1} \]
Geometric Progression
A geometric progression is a sequence where each term is derived from the previous one by multiplying a fixed, non-zero number called the common ratio.
This contrasts with arithmetic sequences, where terms get derived by addition.
Understanding this is vital for differentiating between sequence types and applying the correct formulas and methods of calculation.
Key characteristics include:
This contrasts with arithmetic sequences, where terms get derived by addition.
Understanding this is vital for differentiating between sequence types and applying the correct formulas and methods of calculation.
Key characteristics include:
- Non-zero common ratio: No term of the sequence is zero when the ratio is non-zero.
- Consistent pattern: Once identified, the common ratio provides a consistent pattern to predict subsequent numbers.
- Variation in direction: Sign of the common ratio will affect whether the sequence alternates direction, as seen in \(162, -54, 18, -6...\)
Sequence Calculation
Calculating sequence terms using the nth term formula is both efficient and time-saving.
Once you have the formula, you can calculate any term directly, as demonstrated in the exercise.
Here, brief steps outline the process:
Once you have the formula, you can calculate any term directly, as demonstrated in the exercise.
Here, brief steps outline the process:
- Identify the term needed, such as the 5th or 8th term.
- Substitute this term number into the nth term formula to find the value.
- Use \( n = 5 \) in the formula \( a_n = 162 \left(-\frac{1}{3}\right)^{n-1} \) to get \( a_5 = 2 \).
- Substitute \( n = 8 \), the calculation becomes more complex but is efficiently managed using the nth term formula, resulting in \( a_8 = -\frac{2}{27} \).
