To find the missing numbers in the pattern: ___, 6.84, 7.09, 7.34, ___, 7.84, we first identify the differences between the known numbers.
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Difference between 6.84 and 7.09: \[ 7.09 - 6.84 = 0.25 \]
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Difference between 7.09 and 7.34: \[ 7.34 - 7.09 = 0.25 \]
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Next, we can predict the difference from 7.34 to the next number denoted as \( x \): \[ x - 7.34 = 0.25 \implies x = 7.34 + 0.25 = 7.59 \]
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Finally, we check the difference from \( x \) to 7.84: \[ 7.84 - 7.59 = 0.25 \]
Now that we've filled in the first missing number as 7.59, we now find the first number in the sequence (denoted as \( y \)):
- Calculate the first term using the same difference: \[ 6.84 - y = 0.25 \implies y = 6.84 - 0.25 = 6.59 \]
So the filled pattern is: 6.59, 6.84, 7.09, 7.34, 7.59, 7.84
Thus, the missing numbers are 6.59 and 7.59.