To find an exponential function that represents the geometric sequence \(49, 160, 640, \ldots\), we first need to identify the common ratio of the geometric sequence and then express the terms in a general form.
- Identify the first term (\(a\)) and common ratio (\(r\)):
- The first term \(a = 49\).
- To find the common ratio \(r\), we can divide the second term by the first term: \[ r = \frac{160}{49} \approx 3.2653 \]
- Next, we check the ratio between the third term and the second term to confirm if it is constant: \[ r = \frac{640}{160} = 4 \]
Clearly, the ratios do not remain constant, which indicates that this is not a standard geometric sequence based purely on a common ratio. Let's check the ratios again for clarity:
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From 49 to 160: \[ r = \frac{160}{49} \approx 3.2653 \]
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From 160 to 640: \[ r = \frac{640}{160} = 4 \]
The sequence appears to have varying ratios.
- Find the relationship: To express this sequence in a more general exponential form, we can observe that if we look for a geometric pattern, we can write terms in powers of a variable multiplied by a base number or a factor.
Given the observation, let's consider adjusting our view of the function.
The function \(a_n\) can be expressed in exponential form as:
\[ a_n = a \cdot r^{n-1} \]
However, since the ratio is changing, we'll need a different approach to find the individual terms specifically via a pattern.
We can also express the sequence using \(n\) as the term index:
- \(a_1 = 49 = 7^2\)
- \(a_2 = 160 = 16 \times 10 = 2^4 \times 10\)
- \(a_3 = 640 = 64 \times 10 = 2^6 \times 10\)
Given that upon re-analysis, it is observed that combining powers of 2 and multiples could give a clearer picture.
- Using exponential function: The pattern clearly doesn't follow simple exponential growth but is more a modification. If we choose a formula that captures the variation:
To express \(n\)th term in a function form:
\[ a_n = a \cdot b^{(n-1)} \] might not hold due to changing ratios, but you can also verify if different bases or separating out components in consistent factors can help.
- Final Function:
Let’s summarize:
If we consider that there is no consistent geometric ratio, we can speculate that non-linear functions might represent these sequences. The simplest representation would be:
\[ f(n) = c \cdot k^{n-1} + d \]
You could fit the given values into polynomial or iterative functions.
It's evident that you may want to fully resolve via method like polynomial fitting or another representation dependent on changing sequences.
If the terms still need a simple exponential view, we could fit it into a more composite factor:
Most likely outputting complex forms with coefficients fitting: \[ a_n = 49 \times 4^{(n-1)} \text{ (for approximation)} , \text{(adjusting for nearest behavior)} \]
To summarize, in absence of a true geometric function, breaking into component powers is prudent for analysis. Thus adjusting to confirm or patterns through sequences may strengthen capture factors derived from the terms' behavior or polynomial fits.