In a certain quiz that consists of 10 questions, each question after the first is worth 4 points more than...

1. Translate the problem requirements: We have 10 quiz questions where each question is worth 4 more points than the previous one (arithmetic sequence). The total points for all questions is 360, and we need to find the value of the third question.
2. Set up the arithmetic sequence: Express all question values in terms of the first question's value, using the constant difference of 4 points between consecutive questions.
3. Apply the total constraint: Use the given total of 360 points to create an equation that allows us to solve for the first question's value.
4. Calculate the third question's value: Once we know the first question's value, add the appropriate increments to find the third question's worth.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we know in plain English:

• We have a quiz with 10 questions
• Each question is worth 4 more points than the question before it
• All 10 questions together are worth 360 points
• We need to find how many points the third question is worth

Think of this like climbing stairs where each step is 4 points higher than the previous one. If the first question is worth some amount, the second is worth 4 more, the third is worth 8 more than the first, and so on.

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

2. Set up the arithmetic sequence

Let's call the first question's value 'x' points. Now we can figure out what each question is worth:

• Question 1: x points
• Question 2: x + 4 points (4 more than Question 1)
• Question 3: x + 8 points (4 more than Question 2, or 8 more than Question 1)
• Question 4: x + 12 points
• Question 5: x + 16 points
• Question 6: x + 20 points
• Question 7: x + 24 points
• Question 8: x + 28 points
• Question 9: x + 32 points
• Question 10: x + 36 points

Notice the pattern: each question adds an additional 4 points to the base value x.

3. Apply the total constraint

Now we use the fact that all questions together must equal 360 points. Let's add up all the question values:

\(\mathrm{x + (x + 4) + (x + 8) + (x + 12) + (x + 16) + (x + 20) + (x + 24) + (x + 28) + (x + 32) + (x + 36) = 360}\)

Let's group this smartly:

• We have 10 x's: 10x
• We have the numbers: 4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36

For the numbers, notice they're all multiples of 4: \(\mathrm{4(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)}\)
The sum \(\mathrm{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45}\)
So the sum of our additional points is \(\mathrm{4 \times 45 = 180}\)

Our equation becomes:
\(\mathrm{10x + 180 = 360}\)
\(\mathrm{10x = 180}\)
\(\mathrm{x = 18}\)

So the first question is worth 18 points.

4. Calculate the third question's value

Now that we know the first question is worth 18 points, we can find the third question's value:

Third question = \(\mathrm{x + 8 = 18 + 8 = 26}\) points

Let's verify this makes sense by checking a few values:

• Question 1: 18 points
• Question 2: 18 + 4 = 22 points
• Question 3: 18 + 8 = 26 points ✓

Final Answer

The third question is worth 26 points.

Looking at our answer choices:

1. 18
2. 24
3. 26
4. 32
5. 44

Our answer matches choice C. 26.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting the sequence pattern: Students might confuse "each question after the first is worth 4 points more than the preceding question" and think ALL questions are worth 4 more than the first question, rather than understanding it's a progressive increase where each question builds on the previous one.

2. Setting up the wrong variable: Students may try to set the third question's value as their variable 'x' instead of starting with the first question's value, making the algebra unnecessarily complicated and prone to errors.

3. Misunderstanding what "after the first" means: Some students might include the first question in the 4-point increment pattern, thinking the first question is also 4 points more than something, rather than recognizing it as the base value.

Errors while executing the approach

1. Arithmetic errors in summing the sequence: When calculating 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45, students often make addition mistakes, especially under time pressure, leading to an incorrect total and wrong final answer.

2. Incorrect coefficient counting: Students might miscalculate how many times 'x' appears in the sum (should be 10 times) or forget to factor out the 4 from the additional terms, leading to equations like 10x + 4 + 8 + 12... = 360 instead of 10x + 4(1 + 2 + 3...) = 360.

3. Sign errors or algebraic manipulation mistakes: When solving 10x + 180 = 360, students might subtract incorrectly (getting 10x = 540 instead of 10x = 180) or make division errors when finding x = 18.

Errors while selecting the answer

1. Selecting the first question's value instead of the third: After correctly calculating that x = 18 (the first question's value), students might rush and select answer choice A (18) without completing the final step of finding the third question's value (18 + 8 = 26).

2. Confusing question positions: Students might calculate the wrong question's value due to off-by-one errors, such as finding the second question's value (22 points) or fourth question's value (30 points) instead of the third question's value.