Find the number of three-digit even integers.  with a value greater than 700 but not more than 777

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.

Learn how to find the number of three-digit even integers that are greater than 700 but not more than 777. This problem involves identifying even numbers in a specified range using arithmetic sequence concepts. Suitable for students in Grades 6-8.

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