To find the number of three-digit even integers that are greater than 700 but not more than 777, we can break it down as follows:

1. Range of the numbers: We are interested in three-digit even numbers $n$ such that $700 < n \leq 777$.

2. Even numbers between 700 and 777: The smallest even number greater than 700 is 702, and the largest even number not greater than 777 is 776.

3. Identify the even numbers in this range:

   • The even numbers between 702 and 776 (inclusive) are: 702, 704, 706, 708, ..., 776.

4. Formulate the sequence:

   • This is an arithmetic sequence where the first term $a_1 = 702$, the common difference $d = 2$, and the last term $a_n = 776$.

5. Number of terms in the sequence:

   • The number of terms $n$ in the sequence can be found using the formula for the $n$-th term of an arithmetic sequence: $a_n = a_1 + (n-1) \cdot d$ Substituting the values: $776 = 702 + (n-1) \cdot 2$ Simplifying this: $776 - 702 = (n-1) \cdot 2$ $74 = (n-1) \cdot 2$ $n - 1 = 37$ $n = 38$

Therefore, there are 38 even three-digit integers greater than 700 but not more than 777.

Would you like more details or have any questions? Here are some related questions:

1. How do we determine if a sequence is arithmetic?
2. Can you explain the formula for the $n$-th term of an arithmetic sequence?
3. How would the problem change if the numbers were required to be odd?
4. What if the number needed to be a multiple of 5 instead of even?
5. How would the solution differ if we changed the upper limit to 800?

Tip: Always double-check the constraints (like even/odd or specific ranges) to ensure the correct set of numbers is considered in such problems.