If x is the sum of the first 50 positive even integers and y is the sum of the first...

1. Translate the problem requirements: Clarify what 'first 50 positive even integers' means \((2, 4, 6, \ldots, 100)\) and 'first 50 positive odd integers' means \((1, 3, 5, \ldots, 99)\), then understand we need to find the difference between their sums
2. Recognize the pairing pattern: Instead of calculating each sum separately, identify how even and odd integers relate to each other in a predictable way
3. Apply the difference pattern: Use the consistent relationship between corresponding even and odd numbers to find the total difference without complex calculations
4. Verify with the answer choices: Confirm our result makes sense given the available options

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding exactly what we're looking for. The problem asks us to find two different sums:

• x: the sum of the first 50 positive even integers
• y: the sum of the first 50 positive odd integers

What does "first 50 positive even integers" mean? These are: \(2, 4, 6, 8, 10, \ldots\), and we need exactly 50 of them. To find the 50th even integer, we use the pattern that the nth positive even integer is \(2n\). So the 50th even integer is \(2(50) = 100\).

Therefore: \(x = 2 + 4 + 6 + 8 + \ldots + 100\)

Similarly, "first 50 positive odd integers" means: \(1, 3, 5, 7, 9, \ldots\), and we need exactly 50 of them. The nth positive odd integer is \(2n-1\). So the 50th odd integer is \(2(50)-1 = 99\).

Therefore: \(y = 1 + 3 + 5 + 7 + \ldots + 99\)

We need to find \(x - y\).

Process Skill: TRANSLATE - Converting the problem language into clear mathematical understanding

2. Recognize the pairing pattern

Instead of calculating each sum separately (which would be very time-consuming), let's look for a smart pattern. Notice how even and odd integers relate to each other:

• The 1st even integer is 2, and the 1st odd integer is 1. Difference: \(2 - 1 = 1\)
• The 2nd even integer is 4, and the 2nd odd integer is 3. Difference: \(4 - 3 = 1\)
• The 3rd even integer is 6, and the 3rd odd integer is 5. Difference: \(6 - 5 = 1\)
• The 4th even integer is 8, and the 4th odd integer is 7. Difference: \(8 - 7 = 1\)

Do you see the pattern? Each even integer is exactly 1 more than its corresponding odd integer. This makes sense because consecutive integers differ by 1, and every even integer immediately follows an odd integer.

In general terms: the nth even integer minus the nth odd integer always equals 1.

Process Skill: INFER - Drawing the non-obvious conclusion that we can pair terms rather than calculate separate sums

3. Apply the difference pattern

Now we can use this pattern to find \(x - y\) without doing complex arithmetic:

\(x - y = (2 + 4 + 6 + 8 + \ldots + 100) - (1 + 3 + 5 + 7 + \ldots + 99)\)

We can rearrange this as:

\(x - y = (2 - 1) + (4 - 3) + (6 - 5) + (8 - 7) + \ldots + (100 - 99)\)

Since each pair gives us a difference of 1:

\(x - y = 1 + 1 + 1 + 1 + \ldots + 1\)

How many 1's do we have? We have exactly 50 pairs (since we're pairing the 1st through 50th terms), so:

\(x - y = 50 \times 1 = 50\)

4. Verify with the answer choices

Our calculated answer is 50. Looking at the answer choices:

1. 0
2. 25
3. 50 ✓
4. 75
5. 100

Our answer of 50 corresponds to choice C.

Let's do a quick sanity check with smaller numbers: If we took the first 3 even integers \((2, 4, 6)\) and first 3 odd integers \((1, 3, 5)\):

• Sum of evens: \(2 + 4 + 6 = 12\)
• Sum of odds: \(1 + 3 + 5 = 9\)
• Difference: \(12 - 9 = 3\)

Using our method: 3 pairs, each with difference 1, so total difference = \(3 \times 1 = 3\). ✓

Final Answer

The answer is C. 50

The key insight was recognizing that each even integer is exactly 1 more than its corresponding odd integer, so when we have 50 such pairs, the total difference is \(50 \times 1 = 50\).

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "first 50 positive even integers"

Students often confuse this with "the first 50 multiples of 2" or think it means "all even integers up to 50." They might incorrectly identify the sequence as \(2, 4, 6, \ldots, 50\) instead of \(2, 4, 6, \ldots, 100\). This leads to completely wrong calculations from the start.

2. Attempting to calculate each sum separately using formulas

Many students immediately think to use arithmetic sequence formulas like \(S = n(\text{first term} + \text{last term})/2\) for each sum separately. While this works, it's much more time-consuming and error-prone than recognizing the elegant pairing pattern that makes this problem quick to solve.

3. Not recognizing the pairing opportunity

Students often miss that they can pair corresponding terms (1st even with 1st odd, 2nd even with 2nd odd, etc.) and instead try to find x and y individually. This oversight leads to unnecessary complex calculations and potential arithmetic errors.

Errors while executing the approach

1. Incorrect pairing of terms

Even when students recognize the pairing pattern, they might incorrectly pair terms. For example, pairing 2 with 3, 4 with 5, etc., instead of pairing 2 with 1, 4 with 3, etc. This fundamental misunderstanding of which terms correspond leads to wrong differences.

2. Arithmetic errors in the difference calculation

Students might correctly identify that each pair has a difference of 1 but then make basic multiplication errors, such as calculating \(50 \times 1 = 25\) or getting confused about how many pairs exist (thinking there are 25 pairs instead of 50).

Errors while selecting the answer

1. Calculating \(y - x\) instead of \(x - y\)

Students might correctly find that the absolute difference is 50 but then select -50 if available in the choices, or become confused about the sign. They calculate \((1+3+5+\ldots+99) - (2+4+6+\ldots+100)\) instead of the required \(x - y\), leading them to think the answer is -50 rather than +50.

Alternate Solutions

Smart Numbers Approach

Instead of working with 50 terms directly, let's use smaller numbers to identify the pattern, then scale up.

Step 1: Start with the first 3 terms to identify the pattern

First 3 positive even integers: \(2, 4, 6\)
First 3 positive odd integers: \(1, 3, 5\)
Sum of evens: \(2 + 4 + 6 = 12\)
Sum of odds: \(1 + 3 + 5 = 9\)
Difference: \(12 - 9 = 3\)

Step 2: Check the pattern with 4 terms

First 4 positive even integers: \(2, 4, 6, 8\)
First 4 positive odd integers: \(1, 3, 5, 7\)
Sum of evens: \(2 + 4 + 6 + 8 = 20\)
Sum of odds: \(1 + 3 + 5 + 7 = 16\)
Difference: \(20 - 16 = 4\)

Step 3: Verify the pattern with 5 terms

First 5 positive even integers: \(2, 4, 6, 8, 10\)
First 5 positive odd integers: \(1, 3, 5, 7, 9\)
Sum of evens: \(2 + 4 + 6 + 8 + 10 = 30\)
Sum of odds: \(1 + 3 + 5 + 7 + 9 = 25\)
Difference: \(30 - 25 = 5\)

Step 4: Recognize the clear pattern

• For \(n = 3\) terms: difference = 3
• For \(n = 4\) terms: difference = 4
• For \(n = 5\) terms: difference = 5

The pattern shows that when we take the first n positive even integers minus the first n positive odd integers, the difference equals n.

Step 5: Apply the pattern to our problem

For the first 50 positive even integers minus the first 50 positive odd integers:
Difference = 50

Why this pattern works: Each even integer is exactly 1 more than its corresponding odd integer \((2-1=1, 4-3=1, 6-5=1, \text{etc.})\), so when we have n pairs, the total difference is \(n \times 1 = n\).