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

1. Translate the problem requirements: We have a geometric sequence where each term equals half of the previous term. We know the 10th term falls between 0.0001 and 0.001, and we need to find the range for the 12th term.
2. Establish the relationship pattern: Determine how terms relate to each other in this halving sequence, specifically how the 12th term relates to the 10th term.
3. Apply the halving pattern: Use the fact that going from the 10th to 12th term requires two halving operations to transform the given range.
4. Calculate the resulting range: Apply the transformation to both bounds of the 10th term's range to find the 12th term's range.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in everyday language:

We have a sequence where each number is exactly half of the number before it. Think of it like cutting a piece of paper in half, then cutting that half in half again, and so on.

We're told that the 10th term (the 10th number in our sequence) falls somewhere between 0.0001 and 0.001. We need to figure out what range the 12th term falls in.

In mathematical terms: We have a geometric sequence with common ratio \(\mathrm{r = \frac{1}{2}}\), where \(\mathrm{0.0001 < a_{10} < 0.001}\), and we need to find the range for \(\mathrm{a_{12}}\).

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical relationship

2. Establish the relationship pattern

Since each term is half the previous term, let's trace what happens:

• From the 10th term to the 11th term: we divide by 2 (or multiply by \(\mathrm{\frac{1}{2}}\))
• From the 11th term to the 12th term: we divide by 2 again

So to get from the 10th term directly to the 12th term, we need to apply this halving process twice.

This means: 12th term = (10th term) × \(\mathrm{\frac{1}{2}}\) × \(\mathrm{\frac{1}{2}}\) = (10th term) × \(\mathrm{\frac{1}{4}}\)

In other words, the 12th term is exactly one-quarter of the 10th term.

Process Skill: INFER - Recognizing that we need two halving operations to go from 10th to 12th term

3. Apply the halving pattern

Now we know that the 12th term equals the 10th term divided by 4.

Since the 10th term is between 0.0001 and 0.001, we need to divide both of these boundary values by 4:

• Lower bound: 0.0001 ÷ 4 = 0.000025
• Upper bound: 0.001 ÷ 4 = 0.00025

So the 12th term must be between 0.000025 and 0.00025.

4. Calculate the resulting range

Let's double-check our arithmetic:

For the lower bound:

0.0001 ÷ 4 = 0.0001 ÷ 4 = 0.000025 ✓

For the upper bound:

0.001 ÷ 4 = 0.001 ÷ 4 = 0.00025 ✓

Therefore, the 12th term is between 0.000025 and 0.00025.

Looking at our answer choices, this matches exactly with choice C: 0.000025 and 0.00025.

Final Answer

The 12th term of the sequence is between 0.000025 and 0.00025.

Answer: C

Verification: Since each term is half the previous term, going from the 10th to the 12th term requires two halving operations, which is equivalent to dividing by 4. Applying this to the given range for the 10th term (0.0001 to 0.001) gives us the range 0.000025 to 0.00025 for the 12th term.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the sequence relationship
Students might misinterpret "each term after the first term is one-half the previous term" as meaning the term is doubled rather than halved. This fundamental misunderstanding would lead them to think terms are getting larger instead of smaller, completely derailing their approach.

2. Confusion about term positioning
Students may get confused about how many steps it takes to go from the 10th term to the 12th term. Some might think it's just one step (dividing by 2 once) instead of recognizing that going from term 10 → term 11 → term 12 requires two halving operations.

3. Attempting to find the first term unnecessarily
Some students might think they need to work backwards to find the first term of the sequence before they can find the 12th term. This overcomplicates the problem since we can directly relate the 10th and 12th terms without knowing the first term.

Errors while executing the approach

1. Arithmetic errors with decimal division
When dividing 0.0001 ÷ 4 or 0.001 ÷ 4, students often make errors with decimal place counting. They might get 0.0025 instead of 0.00025, or 0.0000025 instead of 0.000025, shifting decimal places incorrectly.

2. Mixing up which boundary gets which calculation
Students might correctly identify that they need to divide by 4, but then apply the division incorrectly to the bounds - for example, dividing the upper bound by 4 but forgetting to divide the lower bound, or vice versa.

3. Using the wrong divisor
Even if students understand they need to go from the 10th to 12th term, they might use the wrong multiplier. Some might divide by 2 (thinking it's just one step) or divide by 8 (thinking it's three steps), instead of correctly dividing by 4.

Errors while selecting the answer

1. Misreading decimal places in answer choices
Given that all answer choices contain very small decimal numbers with multiple zeros, students often miscount the number of zeros when comparing their calculated answer to the choices. They might select choice B (0.00025 and 0.0025) thinking it matches their answer of 0.000025 to 0.00025.

2. Selecting the range for the wrong term
Some students might correctly calculate a range but then mistakenly select the answer choice that represents the 10th term range instead of the 12th term range, especially if they worked backwards in their calculations and got confused about which range corresponds to which term.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose a smart number for the 10th term

Since the 10th term must be between 0.0001 and 0.001, let's choose a convenient value within this range. We'll use 0.0004 as our smart number because:

• It falls comfortably within the given range
• It's easy to halve repeatedly (0.0004 = 4 × 10⁻⁴)
• The arithmetic will be clean and manageable

Step 2: Apply the sequence rule to find the 12th term

In this sequence, each term is half the previous term, so:

• 11th term = \(\mathrm{\frac{1}{2}}\) × 10th term = \(\mathrm{\frac{1}{2}}\) × 0.0004 = 0.0002
• 12th term = \(\mathrm{\frac{1}{2}}\) × 11th term = \(\mathrm{\frac{1}{2}}\) × 0.0002 = 0.0001

Step 3: Verify this falls within one of the answer ranges

Our calculated 12th term is 0.0001, which falls within the range 0.000025 and 0.00025 (Choice C), since:

0.000025 < 0.0001 < 0.00025

Step 4: Confirm with boundary values

To ensure our answer is robust, let's quickly check what happens at the boundaries:

• If 10th term = 0.0001, then 12th term = 0.0001 ÷ 4 = 0.000025
• If 10th term = 0.001, then 12th term = 0.001 ÷ 4 = 0.00025

This confirms that the 12th term ranges from 0.000025 to 0.00025, which matches Choice C exactly.