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 a_n=a_n-1^r for the geometric sequence
a_1=___
a_n=a_(n-1)*___
write an explicit formula (a_n=a_1^r^(n-1) ) for the geometric sequence. use your explicit formula to find the 32nd term of the sequence
a_n=___*___^(n-1)
a_32=___*___^(__-1)
a_32=___
explain what makes it geometric
it is geometric because you are ___ ___ each time
write a recursive formula a_n=a_n-1^r for the geometric sequence
a_1=___
a_n=a_(n-1)*___
write an explicit formula (a_n=a_1^r^(n-1) ) for the geometric sequence. use your explicit formula to find the 32nd term of the sequence
a_n=___*___^(n-1)
a_32=___*___^(__-1)
a_32=___
Answers
GPT-4o mini
To determine if the sequence \(3, 6, 12, 24, 48, 96\) is geometric, we need to observe the pattern of how each term relates to the previous terms.
**Step 1: Verify it's geometric**
To check if this sequence is geometric, we calculate the ratio of consecutive terms:
- \( \frac{6}{3} = 2 \)
- \( \frac{12}{6} = 2 \)
- \( \frac{24}{12} = 2 \)
- \( \frac{48}{24} = 2 \)
- \( \frac{96}{48} = 2 \)
Since the ratio is constant at \(2\), the sequence is geometric because you are **multiplying by 2 each time**.
**Step 2: Write a recursive formula**
For a geometric sequence, we can define the recursive formula as follows:
- \( a_1 = 3 \)
- \( a_n = a_{n-1} \cdot r \)
Where \( r = 2 \).
So, the recursive formula looks like this:
- \( a_n = a_{n-1} \cdot 2 \)
**Step 3: Write an explicit formula**
The explicit formula for a geometric sequence is given by:
- \( a_n = a_1 \cdot r^{(n-1)} \)
Substituting in our values, we have:
- \( a_1 = 3 \)
- \( r = 2 \)
Thus the explicit formula is:
- \( a_n = 3 \cdot 2^{(n-1)} \)
**Step 4: Calculate the 32nd term**
To find the 32nd term (\(a_{32}\)), we apply the explicit formula:
- \( a_{32} = 3 \cdot 2^{(32 - 1)} = 3 \cdot 2^{31} \)
We need to compute \(2^{31}\):
- \( 2^{31} = 2147483648 \)
Now, we calculate \( a_{32} \):
- \( a_{32} = 3 \cdot 2147483648 = 6442450944 \)
Finally, summarizing:
- Explicit formula: \( a_n = 3 \cdot 2^{(n-1)} \)
- The 32nd term: \( a_{32} = 6442450944 \)
**Step 1: Verify it's geometric**
To check if this sequence is geometric, we calculate the ratio of consecutive terms:
- \( \frac{6}{3} = 2 \)
- \( \frac{12}{6} = 2 \)
- \( \frac{24}{12} = 2 \)
- \( \frac{48}{24} = 2 \)
- \( \frac{96}{48} = 2 \)
Since the ratio is constant at \(2\), the sequence is geometric because you are **multiplying by 2 each time**.
**Step 2: Write a recursive formula**
For a geometric sequence, we can define the recursive formula as follows:
- \( a_1 = 3 \)
- \( a_n = a_{n-1} \cdot r \)
Where \( r = 2 \).
So, the recursive formula looks like this:
- \( a_n = a_{n-1} \cdot 2 \)
**Step 3: Write an explicit formula**
The explicit formula for a geometric sequence is given by:
- \( a_n = a_1 \cdot r^{(n-1)} \)
Substituting in our values, we have:
- \( a_1 = 3 \)
- \( r = 2 \)
Thus the explicit formula is:
- \( a_n = 3 \cdot 2^{(n-1)} \)
**Step 4: Calculate the 32nd term**
To find the 32nd term (\(a_{32}\)), we apply the explicit formula:
- \( a_{32} = 3 \cdot 2^{(32 - 1)} = 3 \cdot 2^{31} \)
We need to compute \(2^{31}\):
- \( 2^{31} = 2147483648 \)
Now, we calculate \( a_{32} \):
- \( a_{32} = 3 \cdot 2147483648 = 6442450944 \)
Finally, summarizing:
- Explicit formula: \( a_n = 3 \cdot 2^{(n-1)} \)
- The 32nd term: \( a_{32} = 6442450944 \)