Answer
A **geometric sequence** is a list of numbers where each term after the first is found by multiplying the previous term by a constant called the **common ratio**. The general formula to find the \(n\)-th term is \(a_n = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term and \(r\) is the common ratio.
Solution
A **geometric sequence** (also known as a geometric progression) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the **common ratio**. This type of sequence grows or decreases exponentially, depending on the value of the common ratio.
### **Key Components**
1. **First Term (\(a_1\))**: The initial term of the sequence.
2. **Common Ratio (\(r\))**: The factor by which consecutive terms are multiplied.
### **General Formula**
The \(n\)-th term (\(a_n\)) of a geometric sequence can be calculated using the formula:
\[
a_n = a_1 \times r^{(n-1)}
\]
Where:
- \(a_n\) = the \(n\)-th term
- \(a_1\) = the first term
- \(r\) = common ratio
- \(n\) = term number
### **Example**
Consider the geometric sequence: 3, 6, 12, 24, 48,...
- **First Term (\(a_1\))**: 3
- **Common Ratio (\(r\))**: 2 (since each term is multiplied by 2 to get the next term)
Using the general formula, the 5th term (\(a_5\)) would be:
\[
a_5 = 3 \times 2^{(5-1)} = 3 \times 16 = 48
\]
### **Sum of a Geometric Sequence**
If you want to find the sum of the first \(n\) terms of a geometric sequence, you can use the following formula (provided \(r \neq 1\)):
\[
S_n = a_1 \times \frac{1 - r^n}{1 - r}
\]
### **Applications**
Geometric sequences are found in various real-life scenarios, such as:
- **Population Growth**: When a population increases by a fixed percentage each year.
- **Finance**: Compound interest calculations where the amount grows by a certain rate each period.
- **Physics**: Processes that involve exponential decay, like radioactive decay.
### **Comparison with Arithmetic Sequences**
Unlike geometric sequences, **arithmetic sequences** have a constant difference between consecutive terms instead of a constant ratio. For example, in the arithmetic sequence 2, 5, 8, 11,... the common difference is 3.
### **Visual Representation**
Graphing a geometric sequence typically results in an exponential curve, increasing or decreasing rapidly based on the common ratio:
- If \(|r| > 1\), the sequence grows exponentially.
- If \(|r| < 1\), the sequence approaches zero exponentially.
- If \(r = 1\), all terms are the same.
- If \(r = -1\), the sequence alternates between two values.
### **Conclusion**
A geometric sequence is a fundamental concept in mathematics that illustrates exponential growth or decay through a constant multiplicative relationship between consecutive terms. Understanding geometric sequences is essential in various fields, including finance, biology, physics, and computer science.
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain
Simplify this solution
