To approximate \( \sqrt{219} \), we can use a function \( f(x) = x^2 \) to evaluate values of \( x \) that get us close to 219. We want to pick \( x \)-values that are near the actual square root of 219.
Calculating \( \sqrt{219} \) gives about 14.8. Therefore, we want to find sequences of \( x \)-values that are centered around this approximation.
Let's analyze the provided sequences:
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4, 4.1, 4.12, 4.123, 4.1231, 4.12311
- These \( x \)-values are much too low and centered around 4.
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16, 21.1, 20.53, 20.521, 20.5191, 20.51916
- The values start around 16, which is too low, but around 20-21, they lie closer to the square root value. However, they are still quite distant from our target of around 14.8.
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4, 4.4, 4.36, 4.359, 4.3589, 4.35890
- These values are again way too low.
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3, 3.9, 3.87, 3.873, 3.8730, 3.87298
- These are much lower than our target again.
Looking more closely, sequence 16, 21.1, 20.53, 20.521, 20.5191, 20.51916 has x-values that are relatively close to where \( \sqrt{219} \approx 14.8 \).
Although 16 is the lowest one in this sequence, the next numbers converge downwards to values that can be narrowed to estimate the square root more closely.
Thus, the best answer is:
16, 21.1, 20.53, 20.521, 20.5191, 20.51916