In a certain sequence, each term after the first term is one-half the previous term. If the...

Question

In a certain sequence, each term after the first term is one-half the previous term. If the tenth term of the sequence is between 0.0001 and 0.001 , then the twelfth term of the sequence is between

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Option C: between 0.000025 and 0.00025 We define the tenth term in terms of the first term of the sequence. a_10 = a_1 (1/2)^9 We then express the twelfth term in terms of the tenth term using the halving rule twice. a_12 = a_10 (1/2)^2 = a_10 1/4 We translate the given range for the tenth term into an inequality. 0.0001 < a_10 < 0.001 Since multiplying an inequality by a positive number preserves its direction, we multiply through by 1/4 to find the range for the twelfth term. 0.0001 < a_10 < 0.001 0.0001 1/4 < a_10 1/4 < 0.001 1/4 0.000025 < a_12 < 0.00025

Explanation	Calculations	Help
$We define the tenth term in terms of the first term of the sequence.$	$a_{10} = a_{1} \times (\frac{1}{2})^{9}$
$We then express the twelfth term in terms of the tenth term using the halving rule twice.$	$a_{12} = a_{10} \times (\frac{1}{2})^{2} = a_{10} \times \frac{1}{4}$
$We translate the given range for the tenth term into an inequality.$	$0.0001 < a_{10} < 0.001$
$Since multiplying an inequality by a positive number preserves its direction, we multiply through by \frac{1}{4} to find the range for the twelfth term.$	$0.0001 < a_{10} < 0.001 0.0001 \times \frac{1}{4} < a_{10} \times \frac{1}{4} < 0.001 \times \frac{1}{4} 0.000025 < a_{12} < 0.00025$

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