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: \( a_{32} = 6442450944 \)
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