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Magazine Chapter 9: Problem 3
Find the \(n\) th term, the fifth term, and the eighth term of the geometric sequence. $$8,4,2,1, \dots$$
Short Answer
Expert verified
n-th term: \( 8 \cdot \left(\frac{1}{2}\right)^{n-1} \); Fifth term: 0.5; Eighth term: 0.0625.
Step by step solution
01
Understand the Sequence
A geometric sequence is a sequence 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 sequence, the first term is 8.
02
Determine the Common Ratio
The common ratio \( r \) is found by dividing the second term by the first term. Here, \( r = \frac{4}{8} = \frac{1}{2} \).
03
Formulate the General Term Formula
The \( n \)th term of a geometric sequence can be found using the formula \( a_n = a_1 \cdot r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.
04
Calculate the n-th Term Formula
Substitute the known values into the formula from Step 3: \( a_n = 8 \cdot \left(\frac{1}{2}\right)^{n-1} \). This is the formula for the \( n \)th term of the sequence.
05
Find the Fifth Term
Use the formula from Step 4 to find the fifth term: \( a_5 = 8 \cdot \left(\frac{1}{2}\right)^{5-1} = 8 \cdot \left(\frac{1}{2}\right)^4 = 8 \cdot \frac{1}{16} = 0.5 \).
06
Find the Eighth Term
Use the formula from Step 4 to find the eighth term: \( a_8 = 8 \cdot \left(\frac{1}{2}\right)^{8-1} = 8 \cdot \left(\frac{1}{2}\right)^7 = 8 \cdot \frac{1}{128} = 0.0625 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Ratio
In a geometric sequence, the common ratio is a crucial element that dictates how the sequence progresses. It is a fixed, constant number that you multiply by the current term to get the next term in the sequence.
For example, in the sequence given:
For example, in the sequence given:
- First term: 8
- Second term: 4
- Common ratio, \( r = \frac{4}{8} = \frac{1}{2} \)
Nth Term Formula
The nth term formula of a geometric sequence lets you find any term in the sequence without having to calculate all the preceding terms. The formula is given by the expression:
- \( a_n = a_1 \cdot r^{n-1} \)
- \( a_1 \) is the first term of the sequence,
- \( r \) is the common ratio,
- \( n \) is the term number you want to find.
Exponential Decay
The sequence in discussion illustrates a concept called exponential decay. This happens when each term in a sequence is a fixed fraction of the previous term. Here, that fraction is represented by the common ratio \( \frac{1}{2} \).
With exponential decay:
With exponential decay:
- The sequence decreases rapidly at first, but the rate of decrease slows over time.
- Each term is half of the one before, meaning the sequence is halving at each step.
Sequence Analysis
Analyzing a geometric sequence involves understanding its pattern of change and how it behaves over time. By looking at the given sequence, we observe:
- The first few terms: 8, 4, 2, 1
- Each term is reduced by a factor of \( \frac{1}{2} \)
- Progression shows decreasing values heading towards zero, never quite reaching it
