Question
Create your own geometric sequence. List at least six terms of your sequence. Explain what makes it geometric.
13.State the fifth term of your geometric sequence using subscript notation.
14. Write an explicit formula for your geometric sequence. Use your explicit formula to find the 100th term of your sequence - your number may be large, make sure you use correct Scientific Notation.
15. Write a recursive formula to represent your geometric sequence, make sure you identify your first term
16.Graph your geometric sequence using coordinate points
13.State the fifth term of your geometric sequence using subscript notation.
14. Write an explicit formula for your geometric sequence. Use your explicit formula to find the 100th term of your sequence - your number may be large, make sure you use correct Scientific Notation.
15. Write a recursive formula to represent your geometric sequence, make sure you identify your first term
16.Graph your geometric sequence using coordinate points
Answers
GPT-4o mini
### Geometric Sequence
**Sequence:** Let's create a geometric sequence where the first term \( a_1 = 3 \) and the common ratio \( r = 2 \).
**Terms of the sequence:**
1. \( a_1 = 3 \)
2. \( a_2 = 3 \times 2 = 6 \)
3. \( a_3 = 6 \times 2 = 12 \)
4. \( a_4 = 12 \times 2 = 24 \)
5. \( a_5 = 24 \times 2 = 48 \)
6. \( a_6 = 48 \times 2 = 96 \)
**So, the sequence is:** \( 3, 6, 12, 24, 48, 96 \)
### Explanation of Geometric Sequence
A sequence is considered geometric if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our sequence, the common ratio \( r \) is 2, so each term is twice the previous term.
### The Fifth Term in Subscript Notation
The fifth term of the geometric sequence using subscript notation is:
\[ a_5 = 48 \]
### Explicit Formula
The explicit formula for a geometric sequence can be written as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Substituting our values, we have:
\[ a_n = 3 \cdot 2^{(n-1)} \]
**Finding the 100th term:**
To find the 100th term:
\[ a_{100} = 3 \cdot 2^{(100-1)} = 3 \cdot 2^{99} \]
Calculating \( 2^{99} \):
Using a calculator or software:
\[ 2^{99} \approx 6.338253001 \times 10^{29} \]
Thus,
\[ a_{100} = 3 \cdot 2^{99} \approx 3.5.622759003 \times 10^{30} \]
### Recursive Formula
The recursive formula for the geometric sequence is:
1. \( a_1 = 3 \)
2. \( a_n = a_{n-1} \cdot 2 \) for \( n > 1 \)
### Graphing the Geometric Sequence
To graph the geometric sequence, we can use the coordinates based on the terms we've calculated. The points would be:
\[
(1, 3), (2, 6), (3, 12), (4, 24), (5, 48), (6, 96)
\]
To create the graph, plot these points on a coordinate system where the x-axis represents the term number (n) and the y-axis represents the term value (\( a_n \)):
- Plot point (1, 3)
- Plot point (2, 6)
- Plot point (3, 12)
- Plot point (4, 24)
- Plot point (5, 48)
- Plot point (6, 96)
You will observe an exponential growth pattern, characteristic of geometric sequences.
**Sequence:** Let's create a geometric sequence where the first term \( a_1 = 3 \) and the common ratio \( r = 2 \).
**Terms of the sequence:**
1. \( a_1 = 3 \)
2. \( a_2 = 3 \times 2 = 6 \)
3. \( a_3 = 6 \times 2 = 12 \)
4. \( a_4 = 12 \times 2 = 24 \)
5. \( a_5 = 24 \times 2 = 48 \)
6. \( a_6 = 48 \times 2 = 96 \)
**So, the sequence is:** \( 3, 6, 12, 24, 48, 96 \)
### Explanation of Geometric Sequence
A sequence is considered geometric if each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our sequence, the common ratio \( r \) is 2, so each term is twice the previous term.
### The Fifth Term in Subscript Notation
The fifth term of the geometric sequence using subscript notation is:
\[ a_5 = 48 \]
### Explicit Formula
The explicit formula for a geometric sequence can be written as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Substituting our values, we have:
\[ a_n = 3 \cdot 2^{(n-1)} \]
**Finding the 100th term:**
To find the 100th term:
\[ a_{100} = 3 \cdot 2^{(100-1)} = 3 \cdot 2^{99} \]
Calculating \( 2^{99} \):
Using a calculator or software:
\[ 2^{99} \approx 6.338253001 \times 10^{29} \]
Thus,
\[ a_{100} = 3 \cdot 2^{99} \approx 3.5.622759003 \times 10^{30} \]
### Recursive Formula
The recursive formula for the geometric sequence is:
1. \( a_1 = 3 \)
2. \( a_n = a_{n-1} \cdot 2 \) for \( n > 1 \)
### Graphing the Geometric Sequence
To graph the geometric sequence, we can use the coordinates based on the terms we've calculated. The points would be:
\[
(1, 3), (2, 6), (3, 12), (4, 24), (5, 48), (6, 96)
\]
To create the graph, plot these points on a coordinate system where the x-axis represents the term number (n) and the y-axis represents the term value (\( a_n \)):
- Plot point (1, 3)
- Plot point (2, 6)
- Plot point (3, 12)
- Plot point (4, 24)
- Plot point (5, 48)
- Plot point (6, 96)
You will observe an exponential growth pattern, characteristic of geometric sequences.