Anonymous
2 hours ago
Fully explain the topic of geometric sequence.
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ChatGPT
2 hours ago
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In other words, each term in a geometric sequence is obtained by multiplying the previous term by a constant factor.
The general form of a geometric sequence is: a, ar, ar^2, ar^3, ..., where 'a' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the sequence.
For example, if the first term of a geometric sequence is 2 and the common ratio is 3, the sequence would be: 2, 6, 18, 54, ...
Geometric sequences have several important properties, including:
1. The nth term of a geometric sequence can be found using the formula: a_n = a * r^(n-1), where 'a_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the sequence.
2. The sum of the first 'n' terms of a geometric sequence can be found using the formula: S_n = a * (1 - r^n) / (1 - r), where 'S_n' is the sum of the first 'n' terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms in the sequence.
Geometric sequences are commonly used in mathematics, finance, and science to model exponential growth or decay. They are also used in various real-world applications, such as population growth, compound interest, and radioactive decay.
The general form of a geometric sequence is: a, ar, ar^2, ar^3, ..., where 'a' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the sequence.
For example, if the first term of a geometric sequence is 2 and the common ratio is 3, the sequence would be: 2, 6, 18, 54, ...
Geometric sequences have several important properties, including:
1. The nth term of a geometric sequence can be found using the formula: a_n = a * r^(n-1), where 'a_n' is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the position of the term in the sequence.
2. The sum of the first 'n' terms of a geometric sequence can be found using the formula: S_n = a * (1 - r^n) / (1 - r), where 'S_n' is the sum of the first 'n' terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms in the sequence.
Geometric sequences are commonly used in mathematics, finance, and science to model exponential growth or decay. They are also used in various real-world applications, such as population growth, compound interest, and radioactive decay.
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Google Gemini AI
2 hours ago
Geometric Sequence
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant factor, known as the common ratio.
Properties
- The general term of a geometric sequence is given by:
```
a_n = a_1 - r^(n-1)
```
Where:
- a_n is the nth term
- a_1 is the first term
- r is the common ratio
- The common ratio can be positive or negative.
Convergence and Divergence
A geometric sequence converges if the common ratio is less than 1 in absolute value (i.e., |r| < 1). In this case, the terms approach zero as n gets large.
A geometric sequence diverges if the common ratio is greater than or equal to 1 in absolute value (i.e., |r| ≥ 1). In this case, the terms either increase or decrease indefinitely.
Sum of a Geometric Sequence
The sum of the first n terms of a geometric sequence is given by:
```
S_n = a_1 - (1 - r^n) / (1 - r)
```
Where |r| < 1.
Applications
Geometric sequences have numerous applications in real-world situations, such as:
- Modeling population growth or decay
- Calculating compound interest
- Analyzing radioactive decay
- Solving financial problems involving annuities
Example
Consider a geometric sequence with the first term a_1 = 2 and the common ratio r = 3.
- The second term is a_2 = 2 - 3 = 6.
- The third term is a_3 = 6 - 3 = 18.
- The nth term is a_n = 2 - 3^(n-1).
The sequence diverges because |r| = 3 is greater than 1.
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a constant factor, known as the common ratio.
Properties
- The general term of a geometric sequence is given by:
```
a_n = a_1 - r^(n-1)
```
Where:
- a_n is the nth term
- a_1 is the first term
- r is the common ratio
- The common ratio can be positive or negative.
Convergence and Divergence
A geometric sequence converges if the common ratio is less than 1 in absolute value (i.e., |r| < 1). In this case, the terms approach zero as n gets large.
A geometric sequence diverges if the common ratio is greater than or equal to 1 in absolute value (i.e., |r| ≥ 1). In this case, the terms either increase or decrease indefinitely.
Sum of a Geometric Sequence
The sum of the first n terms of a geometric sequence is given by:
```
S_n = a_1 - (1 - r^n) / (1 - r)
```
Where |r| < 1.
Applications
Geometric sequences have numerous applications in real-world situations, such as:
- Modeling population growth or decay
- Calculating compound interest
- Analyzing radioactive decay
- Solving financial problems involving annuities
Example
Consider a geometric sequence with the first term a_1 = 2 and the common ratio r = 3.
- The second term is a_2 = 2 - 3 = 6.
- The third term is a_3 = 6 - 3 = 18.
- The nth term is a_n = 2 - 3^(n-1).
The sequence diverges because |r| = 3 is greater than 1.
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