1/2, 1/4, 1/8, 1/16, 1/32, .... In the sequence above each term after after the first one-half the previous term....

1. Translate the problem requirements: We have a sequence where each term is half the previous term, starting with \(\frac{1}{2}\). We need to find the 10th term and determine which inequality it satisfies.
2. Identify the sequence pattern: Recognize this as a geometric sequence and establish the general term formula using the pattern of halving.
3. Calculate the 10th term: Use the pattern to find the exact value of the 10th term without unnecessary complex calculations.
4. Compare with answer choices: Convert the fraction to decimal form and match it against the given ranges to identify the correct inequality.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we're given and what we need to find.

We have a sequence that starts with \(\frac{1}{2}\), and each term after that is half of the previous term. So:

• First term: \(\frac{1}{2}\)
• Second term: half of \(\frac{1}{2}\), which is \(\frac{1}{4}\)
• Third term: half of \(\frac{1}{4}\), which is \(\frac{1}{8}\)
• Fourth term: half of \(\frac{1}{8}\), which is \(\frac{1}{16}\)
• And so on...

We need to find the 10th term in this sequence, call it x, and then determine which of the given inequalities contains this value.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical understanding of what we're looking for

2. Identify the sequence pattern

Let's look at the pattern more carefully by writing out what we see:

• 1st term: \(\frac{1}{2} = \frac{1}{2^1}\)
• 2nd term: \(\frac{1}{4} = \frac{1}{2^2}\)
• 3rd term: \(\frac{1}{8} = \frac{1}{2^3}\)
• 4th term: \(\frac{1}{16} = \frac{1}{2^4}\)
• 5th term: \(\frac{1}{32} = \frac{1}{2^5}\)

Do you see the pattern? Each term can be written as 1 divided by 2 raised to some power. Specifically, the nth term equals \(\frac{1}{2^n}\).

This makes sense because each time we take half of the previous term, we're multiplying by \(\frac{1}{2}\), which is the same as multiplying the denominator by 2.

So our general formula is: \(\mathrm{nth\,term} = \frac{1}{2^n}\)

3. Calculate the 10th term

Using our pattern, the 10th term is:
\(x = \frac{1}{2^{10}}\)

Now we need to calculate what \(2^{10}\) equals. Let's build this up step by step:

• \(2^1 = 2\)
• \(2^2 = 4\)
• \(2^3 = 8\)
• \(2^4 = 16\)
• \(2^5 = 32\)
• \(2^6 = 64\)
• \(2^7 = 128\)
• \(2^8 = 256\)
• \(2^9 = 512\)
• \(2^{10} = 1024\)

Therefore: \(x = \frac{1}{1024}\)

4. Compare with answer choices

Now we need to convert \(\frac{1}{1024}\) to decimal form to see which inequality it satisfies.

To convert \(\frac{1}{1024}\) to decimal, let's think about it in terms of powers of 10:

• \(\frac{1}{10} = 0.1\)
• \(\frac{1}{100} = 0.01\)
• \(\frac{1}{1000} = 0.001\)
• \(\frac{1}{10000} = 0.0001\)

Since 1024 is between 1000 and 10000, we know that \(\frac{1}{1024}\) is between \(\frac{1}{10000}\) and \(\frac{1}{1000}\).

More precisely: \(\frac{1}{1024} \approx 0.0009765625\)

This means: \(0.0001 < \frac{1}{1024} < 0.001\)

Looking at our answer choices:

1. \(0.1 < x < 1\) (too big)
2. \(0.01 < x < 0.1\) (too big)
3. \(0.001 < x < 0.01\) (too big)
4. \(0.0001 < x < 0.001\) (this matches!)
5. \(0.00001 < x < 0.0001\) (too small)

Final Answer

The 10th term of the sequence is \(x = \frac{1}{1024} \approx 0.000977\), which satisfies the inequality \(0.0001 < x < 0.001\).

The correct answer is D.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the sequence starting point: Students may incorrectly assume the sequence starts from the "first term" being \(\frac{1}{2}\), but then confuse whether \(\frac{1}{2}\) is the 0th term or 1st term when applying the formula. This leads to using n-1 in the formula instead of n, resulting in calculating the 9th term instead of the 10th term.

2. Incorrect pattern identification: Students might incorrectly identify the pattern as multiplying by \(\frac{1}{2}\) each time and write the nth term as \(\frac{1}{2} \times (\frac{1}{2})^{n-1}\) instead of recognizing that the nth term is simply \((\frac{1}{2})^n\). This confusion stems from overthinking the "each term after the first" instruction.

Errors while executing the approach

1. Calculation errors with powers of 2: Students often make arithmetic mistakes when calculating \(2^{10}\), either by miscounting the repeated multiplications or by confusing powers of 2. Common incorrect values include 512 (which is \(2^9\)) or 2048 (which is \(2^{11}\)).

2. Decimal conversion errors: When converting \(\frac{1}{1024}\) to decimal form, students may make computational errors, especially when doing long division manually, or may incorrectly estimate the decimal value without proper calculation.

3. Inequality comparison mistakes: Students may correctly calculate \(\frac{1}{1024} \approx 0.000977\) but then make errors when comparing this value to the given ranges, particularly confusing which boundary values to use or making errors with the number of zeros in the decimal representations.

Errors while selecting the answer

1. Boundary confusion: Students might correctly calculate that \(x \approx 0.000977\) but then select choice C (\(0.001 < x < 0.01\)) because they incorrectly think \(0.000977 > 0.001\), when in fact \(0.000977 < 0.001\). This error often occurs due to hasty comparison of decimal values with many zeros.