To find the number of three-digit odd integers that are greater than 700 but not more than 777, we can follow these steps:

1. Range of the numbers: We need to identify odd numbers $n$ such that $700 < n \leq 777$.

2. Odd numbers between 700 and 777: The smallest odd number greater than 700 is 701, and the largest odd number not greater than 777 is 777.

3. List the sequence:

   • The odd numbers between 701 and 777 (inclusive) are: 701, 703, 705, 707, ..., 777.

4. Formulate the sequence:

   • This is an arithmetic sequence where the first term $a_1 = 701$, the common difference $d = 2$, and the last term $a_n = 777$.

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: $777 = 701 + (n-1) \cdot 2$ Simplifying this: $777 - 701 = (n-1) \cdot 2$ $76 = (n-1) \cdot 2$ $n - 1 = 38$ $n = 39$

Therefore, there are 39 odd 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 arithmetic sequences differ from geometric sequences?
2. What is the formula to find the sum of an arithmetic sequence?
3. How would the problem change if the numbers were required to be even?
4. What if the lower limit was changed to 600 but still not more than 777?
5. Can you explain how to find the sum of the odd numbers within this range?

Tip: When dealing with sequences, always confirm the type (arithmetic, geometric, etc.) and the relevant formula to ensure accurate calculations.