In the sequence x_0, x_1, x_2, ldots, x_n, each term from x_1 to x_k is 3 greater than the previous...

1. Translate the problem requirements: We have a sequence that starts at 0, increases by 3 each step until position k where it reaches 15, then decreases by 3 each step until it returns to 0 at position n. We need to find the total length n.
2. Determine the rising phase structure: Calculate how many steps it takes to go from \(\mathrm{x_0 = 0}\) to \(\mathrm{x_k = 15}\), increasing by 3 each time.
3. Determine the falling phase structure: Calculate how many steps it takes to go from \(\mathrm{x_k = 15}\) back down to \(\mathrm{x_n = 0}\), decreasing by 3 each time.
4. Calculate total sequence length: Add the rising steps and falling steps to find n, the final position in the sequence.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this sequence in everyday language:

We start at position 0 with a value of 0. Then we keep adding 3 to get the next term, so we go: 0, 3, 6, 9, 12, 15, ... until we reach position k where the value is 15.

After reaching 15 at position k, we change direction and start subtracting 3 from each term: 15, 12, 9, 6, 3, 0, until we get back to 0 at the final position n.

Think of this like climbing up a mountain (adding 3 each step) until you reach the peak at 15, then walking back down the same mountain (subtracting 3 each step) until you're back at ground level.

Process Skill: TRANSLATE - Converting the sequence description into a clear mental picture

2. Determine the rising phase structure

Now let's figure out how many steps it takes to climb from 0 to 15, adding 3 each time:

Starting value: \(\mathrm{x_0 = 0}\)
After 1 step: \(\mathrm{x_1 = 0 + 3 = 3}\)
After 2 steps: \(\mathrm{x_2 = 3 + 3 = 6}\)
After 3 steps: \(\mathrm{x_3 = 6 + 3 = 9}\)
After 4 steps: \(\mathrm{x_4 = 9 + 3 = 12}\)
After 5 steps: \(\mathrm{x_5 = 12 + 3 = 15}\)

So it takes exactly 5 steps to go from 0 to 15, which means k = 5.

We can also think about this mathematically: if we add 3 each time and need to reach 15, then \(\mathrm{k = 15 ÷ 3 = 5}\).

3. Determine the falling phase structure

Now let's figure out how many steps it takes to come back down from 15 to 0, subtracting 3 each time:

Starting from position k where \(\mathrm{x_k = 15}\):
After 1 step down: \(\mathrm{x_{k+1} = 15 - 3 = 12}\)
After 2 steps down: \(\mathrm{x_{k+2} = 12 - 3 = 9}\)
After 3 steps down: \(\mathrm{x_{k+3} = 9 - 3 = 6}\)
After 4 steps down: \(\mathrm{x_{k+4} = 6 - 3 = 3}\)
After 5 steps down: \(\mathrm{x_{k+5} = 3 - 3 = 0}\)

So it takes exactly 5 steps to go from 15 back down to 0.

This makes sense because we're essentially retracing our steps - the same number of steps up equals the same number of steps down.

4. Calculate total sequence length

Now we can find the total length of our sequence:

We know that:
• It takes 5 steps to go from position 0 to position k (where k = 5)
• It takes 5 more steps to go from position k to position n

So the total number of positions from start to finish is:
n = steps up + steps down = 5 + 5 = 10

Let's verify this makes sense:
Position 0: value = 0
Positions 1-5: values = 3, 6, 9, 12, 15 (going up)
Positions 6-10: values = 12, 9, 6, 3, 0 (going down)

So position 5 has value 15 (this matches \(\mathrm{x_k = 15}\) where k = 5)
And position 10 has value 0 (this matches \(\mathrm{x_n = 0}\) where n = 10)

Process Skill: VISUALIZE - Seeing the complete sequence structure to verify our answer

Final Answer

The value of n is 10, which corresponds to answer choice (D).

Our sequence has 11 terms total (from position 0 to position 10), where we climb for 5 steps to reach the peak value of 15, then descend for 5 steps to return to 0.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the sequence structure
Students often misinterpret the problem as having a linear sequence that only increases or only decreases, rather than recognizing it has two distinct phases - an ascending phase followed by a descending phase. They might think the sequence continues adding 3 throughout, or that it starts decreasing from the beginning.

2. Confusion about the indexing and boundary conditions
Students frequently get confused about whether the sequence starts at position 0 or position 1, and whether \(\mathrm{x_0 = 0}\) and \(\mathrm{x_n = 0}\) are included in the counting. They might incorrectly assume the sequence has (n-1) terms instead of (n+1) terms, leading to off-by-one errors in their approach.

3. Misinterpreting the relationship between k and the peak value
Students often fail to recognize that k represents the position where the sequence reaches its maximum value of 15, not the number of steps taken. They might confuse k with the count of increments, leading them to set up incorrect equations for finding the sequence length.

Errors while executing the approach

1. Arithmetic errors in determining k
When calculating how many steps it takes to reach 15 from 0 by adding 3 each time, students might make simple division errors (15 ÷ 3 = 5) or forget to account for the starting position correctly, leading to incorrect values for k.

2. Incorrect symmetry assumption
While the sequence is symmetric in this problem, students might incorrectly assume this symmetry without verifying it, or conversely, they might fail to recognize the symmetry and unnecessarily complicate their calculations by working out the descending phase step-by-step when they could simply double the ascending phase length.

3. Mixing up positions and values
Students often confuse the position indices (0, 1, 2, ..., k, ..., n) with the actual values at those positions (0, 3, 6, ..., 15, ..., 0). This leads to errors when trying to determine relationships between k, n, and the sequence values.

Errors while selecting the answer

1. Selecting k instead of n
After correctly determining that k = 5, students might mistakenly select 5 as their final answer instead of recognizing that the question asks for n, which is the final position in the sequence (n = 10).

2. Off-by-one error in final counting
Students might correctly identify that there are 5 steps up and 5 steps down but then incorrectly conclude that n = 5 + 5 - 1 = 9, thinking they need to subtract 1 to avoid double-counting the peak position, when in fact n = 5 + 5 = 10.