GMAT Practice Question: 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
- 0.0025 and 0.025
- 0.00025 and 0.0025
- 0.000025 and 0.00025
- 0.0000025 and 0.000025
- 0.00000025 and 0.0000025
Topics: sequences & series, decimals, inequalities
Solution
Step 1
We define the tenth term in terms of the first term of the sequence.
a_{10} = a_{1} \times (\frac{1}{2})^9
Step 2
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}
Step 3
We translate the given range for the tenth term into an inequality.
0.0001 < a_{10} < 0.001
Step 4
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
Answer
Option C: between 0.000025 and 0.00025
