To write an exponential function that represents the geometric sequence \(2, 8, 32, 128, 512, 2048, \ldots\), we can use the form of an exponential function. The sequence can be expressed as a function of \(n\).
Step 1: Identify the Exponential Function
Given that the first term \(a_1 = 2\) and the common ratio \(r = 4\), we can write the exponential function in the following form:
\[
f(n) = a \cdot b^{n-1}
\]
where:
- \(a\) is the first term (which is 2),
- \(b\) is the common ratio (which is 4),
- \(n\) is the term index starting from 1.
So the exponential function becomes:
\[
f(n) = 2 \cdot 4^{n-1}
\]
Step 2: Compare with the Explicit Formula
The explicit formula we established previously is:
\[
a_n = 2 \cdot 4^{(n-1)}
\]
This is already in the same form as the exponential function we just derived:
\[
f(n) = 2 \cdot 4^{n-1}
\]
This shows that the exponential function has the same structure as the explicit formula for the geometric sequence.
Step 3: Evaluating for the 100th Term
Now, to find the 100th term using the function, you simply plug \(n = 100\) into the formula:
\[
a_{100} = 2 \cdot 4^{100-1} = 2 \cdot 4^{99}
\]
Step 4: Verify with the Given Format
Your request referred to a format \( a_n = 10(4)^{n-1} \). It appears there is a slight discrepancy since the first term from our sequence is 2, not 10. Based on the original series, if you specifically use:
\[
a_n = 10(4)^{n-1}
\]
then you are actually modifying the sequence because if the first term is meant to be \(10\), the subsequent terms would not match the original sequence of \(2, 8, 32, 128, 512, 2048, \ldots\).
If we take your suggested format seriously, it may look like:
- The numerical factor \(10\) indicates a different sequence entirely.
- In our established sequence, every term multiples by \(4\) (the original form leads to \(2, 8, 32, \ldots\), and maintains the ratio of \(4\)).
Thus, the correct exponential function to represent the original geometric sequence is indeed:
\[
f(n) = 2 \cdot 4^{n-1}
\]
And for \(n = 100\):
\[
a_{100} = 2 \cdot 4^{99}
\]
Conclusion
While the function \(a_n = 10(4)^{n-1}\) does follow the same exponential structure, it does not represent the original sequence provided. The consistent and correct representation remains \(a_n = 2 \cdot 4^{n-1}\).
