To find the correct formula for the sequence \( 2, 3.5, 6.125, 10.71875, \ldots \), we need to identify a pattern or a relationship.
Let's analyze the provided sequence:
- The first term \( a_1 = 2 \).
- The second term \( a_2 = 3.5 \).
- The third term \( a_3 = 6.125 \).
- The fourth term \( a_4 = 10.71875 \).
To find a connection, we can look at the ratios of consecutive terms:
- \( \frac{a_2}{a_1} = \frac{3.5}{2} = 1.75 \)
- \( \frac{a_3}{a_2} = \frac{6.125}{3.5} = 1.75 \)
- \( \frac{a_4}{a_3} = \frac{10.71875}{6.125} = 1.75 \)
The ratio between consecutive terms appears to be consistently \( 1.75 \).
This suggests a geometric sequence where the first term is \( 2 \) and the common ratio is \( 1.75 \).
The general formula for a geometric sequence can be written as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
Where:
- \( a_1 = 2 \)
- \( r = 1.75 \)
Hence, the formula for the \( n \)-th term would be:
\[ y = 2 \cdot (1.75)^{(n-1)} \]
Looking at the provided response options, this corresponds to:
y = 2(1.75)^{x-1}
This is indeed the correct formula for the given sequence.
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