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Magazine Chapter 11: Problem 53
OPEN ENDED Write a geometric sequence with a common ratio of \(\frac{2}{3}\) .
Short Answer
Expert verified
1, \(\frac{2}{3}\), \(\frac{4}{9}\), \(\frac{8}{27}\), ... (common ratio is \(\frac{2}{3}\)).
Step by step solution
01
Understanding Geometric 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. For this problem, the common ratio is \( \frac{2}{3} \).
02
Choosing the First Term
You can choose any non-zero number as the first term of the sequence. Let's choose the first term \( a_1 \) to be 1. This makes it easier to follow the relationship between terms.
03
Calculating the Second Term
The second term \( a_2 \) is found by multiplying the first term by the common ratio. So, \( a_2 = a_1 \times \frac{2}{3} = 1 \times \frac{2}{3} = \frac{2}{3} \).
04
Calculating the Third Term
The third term \( a_3 \) is found by multiplying the second term by the common ratio again. So, \( a_3 = a_2 \times \frac{2}{3} = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9} \).
05
Extending the Sequence
Continue this process to find more terms. The fourth term \( a_4 = a_3 \times \frac{2}{3} = \frac{4}{9} \times \frac{2}{3} = \frac{8}{27} \). You can continue this pattern to find additional terms if needed.
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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 critical component. It is the consistent factor by which each term in the sequence is multiplied to achieve the next term. For instance, if the common ratio is \( \frac{2}{3} \), as given in this exercise, it means that each term of the sequence is obtained by multiplying the previous term by \( \frac{2}{3} \).
Understanding the role of the common ratio helps in quickly identifying the pattern of a sequence. When you observe sequences, the common ratio remains constant, contributing to the predictable nature of geometric progressions.
Key Points of Common Ratio:
Understanding the role of the common ratio helps in quickly identifying the pattern of a sequence. When you observe sequences, the common ratio remains constant, contributing to the predictable nature of geometric progressions.
Key Points of Common Ratio:
- It is a fixed number.
- All terms are affected by this multiplier.
- It can be any non-zero number, including fractions and decimals.
Geometric Progression
Geometric progression refers to a sequence where each term is derived by multiplying the previous one by the common ratio. This kind of sequence is distinct from an arithmetic progression, where terms are added by a constant value.
In a geometric progression:
In a geometric progression:
- Each term is dependent on the preceding term and the common ratio.
- The multiplication can significantly change the number size depending if the common ratio is greater or less than one.
- The sequence can be either increasing, decreasing, or constant, determined by the value of the common ratio.
Sequence Terms
The terms in a geometric sequence are generated systematically. Starting with a chosen first term, each subsequent term is obtained by applying the common ratio repeatedly.
For example, if you start with the term \( a_1 = 1 \) and a common ratio of \( \frac{2}{3} \), the sequence would be:
For example, if you start with the term \( a_1 = 1 \) and a common ratio of \( \frac{2}{3} \), the sequence would be:
- First Term: \( a_1 = 1 \)
- Second Term: \( a_2 = 1 \times \frac{2}{3} = \frac{2}{3} \)
- Third Term: \( a_3 = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9} \)
- Fourth Term: \( a_4 = \frac{4}{9} \times \frac{2}{3} = \frac{8}{27} \)
Mathematics Education
In mathematics education, understanding geometric sequences is fundamental. It provides students with insights into how series and sequences work. Here's how learning about geometric sequences can be beneficial:
Importance of Geometric Sequences in Education:
Importance of Geometric Sequences in Education:
- A building block for more complex mathematical concepts such as calculus and algebra.
- Develops logical reasoning and pattern recognition skills.
- Applicable in many fields, encouraging interdisciplinary learning such as in economics and science.
