To solve this, let us analyze the problem step by step:

Step 1: Formula for the Sum of First $n$ Even Numbers

The sum of the first $n$ even numbers is given by: $S_{\text{even}} = n \cdot (n + 1)$ For $n = 65$, the sum is: $S_{\text{even}} = 65 \cdot (65 + 1) = 65 \cdot 66 = 4290$

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Step 2: Formula for the Sum of First $n$ Odd Numbers

The sum of the first $n$ odd numbers is given by: $S_{\text{odd}} = n^2$ For $n = 65$, the sum is: $S_{\text{odd}} = 65^2 = 4225$

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Step 3: Difference Between the Two Sums

The difference between the sum of the first 65 even numbers and the sum of the first 65 odd numbers is: $\text{Difference} = S_{\text{even}} - S_{\text{odd}} = 4290 - 4225 = 65$

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Final Answer:

The difference is: $\boxed{65}$

Would you like a more detailed explanation or additional examples? Here are 5 related questions to consider:

1. What is the sum of the first 100 even numbers?
2. What is the sum of the first 50 odd numbers?
3. What is the difference between the sums of the first $n$ even and odd numbers for a general $n$?
4. How can you derive the formulas for the sums of even and odd numbers?
5. How does this problem relate to arithmetic sequences?

Tip: Remember that the difference between the sums of the first $n$ even and odd numbers is always $n$, because each even number is exactly 1 more than the corresponding odd number.