Question
Given the sequence 3, 6, 12, 24, 48, 96 …
Explain what makes it geometric.
It is geometric because you are _ _ each time.
Write a recursive formula an = an-1r for the geometric sequence.
a1 + _
an = a(n-1)⋅.
Write an explicit formula (an = a1r^(n-1)) for the geometric sequence. Use your explicit formula to find the 32nd term of the sequence.
an = _⋅_^(n-1)
a32 = _⋅_^(_-1)
a32 = _
Explain what makes it geometric.
It is geometric because you are _ _ each time.
Write a recursive formula an = an-1r for the geometric sequence.
a1 + _
an = a(n-1)⋅.
Write an explicit formula (an = a1r^(n-1)) for the geometric sequence. Use your explicit formula to find the 32nd term of the sequence.
an = _⋅_^(n-1)
a32 = _⋅_^(_-1)
a32 = _
Answers
GPT-4o mini
The given sequence is 3, 6, 12, 24, 48, 96 ...
### What makes it geometric:
It is geometric because you are **multiplying by 2** each time.
### Recursive Formula:
The recursive formula for a geometric sequence is given by:
\[ a_n = a_{n-1} \cdot r \]
For this sequence, we have:
- \( a_1 = 3 \) (the first term)
- The common ratio \( r = 2 \)
So the recursive formula can be written as:
\[ a_n = a_{n-1} \cdot 2 \]
### Explicit Formula:
The explicit formula for a geometric sequence is:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Using the initial term \( a_1 = 3 \) and common ratio \( r = 2 \), we have:
\[ a_n = 3 \cdot 2^{(n-1)} \]
### Finding the 32nd Term:
To find the 32nd term, we substitute \( n = 32 \) into the explicit formula:
\[ a_{32} = 3 \cdot 2^{(32-1)} \]
\[ a_{32} = 3 \cdot 2^{31} \]
### Final Calculation of a32:
Calculating \( 2^{31} \):
1. \( 2^{31} = 2147483648 \)
2. Hence, \( a_{32} = 3 \cdot 2147483648 \)
3. Therefore, \( a_{32} = 6442450944 \)
### Final Answers:
- Recursive formula: \( a_n = a_{n-1} \cdot 2 \)
- Explicit formula: \( a_n = 3 \cdot 2^{(n-1)} \)
- \( a_{32} = 6442450944 \)
### What makes it geometric:
It is geometric because you are **multiplying by 2** each time.
### Recursive Formula:
The recursive formula for a geometric sequence is given by:
\[ a_n = a_{n-1} \cdot r \]
For this sequence, we have:
- \( a_1 = 3 \) (the first term)
- The common ratio \( r = 2 \)
So the recursive formula can be written as:
\[ a_n = a_{n-1} \cdot 2 \]
### Explicit Formula:
The explicit formula for a geometric sequence is:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Using the initial term \( a_1 = 3 \) and common ratio \( r = 2 \), we have:
\[ a_n = 3 \cdot 2^{(n-1)} \]
### Finding the 32nd Term:
To find the 32nd term, we substitute \( n = 32 \) into the explicit formula:
\[ a_{32} = 3 \cdot 2^{(32-1)} \]
\[ a_{32} = 3 \cdot 2^{31} \]
### Final Calculation of a32:
Calculating \( 2^{31} \):
1. \( 2^{31} = 2147483648 \)
2. Hence, \( a_{32} = 3 \cdot 2147483648 \)
3. Therefore, \( a_{32} = 6442450944 \)
### Final Answers:
- Recursive formula: \( a_n = a_{n-1} \cdot 2 \)
- Explicit formula: \( a_n = 3 \cdot 2^{(n-1)} \)
- \( a_{32} = 6442450944 \)