Let `x therefore y` be defined as the sum of all integers between x and y. For example, `1 therefore4=2+3=5`. What is the value of `(60 therefore 900)-(63 therefore 898)`?

Answer

To solve the problem, we need to compute the value of \( (60 \text{ therefore } 900) - (63 \text{ therefore } 898) \).

### Step-by-Step Solution:

1. **Understanding the Operation**:
The operation \( x \text{ therefore } y \) is defined as the sum of all integers between \( x \) and \( y \). This means we sum all integers from \( x + 1 \) to \( y - 1 \).

2. **Calculate \( 60 \text{ therefore } 900 \)**:
\[ 60 \text{ therefore } 900 = 61 + 62 + 63 + \ldots + 899 \]
This is the sum of integers from 61 to 899.

3. **Calculate the number of terms**:
The first term is 61 and the last term is 899. The number of terms can be calculated as:
\[ n = 899 - 61 + 1 = 839 \]

4. **Calculate the sum using the formula for the sum of an arithmetic series**:
The sum \( S \) of the first \( n \) terms of an arithmetic series can be calculated using the formula:
\[ S = \frac{n}{2} \times (\text{first term} + \text{last term}) \]
Applying this:
\[ S = \frac{839}{2} \times (61 + 899) = \frac{839}{2} \times 960 = 839 \times 480 = 403920 \]

5. **Calculate \( 63 \text{ therefore } 898 \)**:
\[ 63 \text{ therefore } 898 = 64 + 65 + 66 + \ldots + 897 \]
This is the sum of integers from 64 to 897.

6. **Calculate the number of terms**:
The first term is 64 and the last term is 897. The number of terms can be calculated as:
\[ n = 897 - 64 + 1 = 834 \]

7. **Calculate the sum using the formula for the sum of an arithmetic series**:
\[ S = \frac{834}{2} \times (64 + 897) = \frac{834}{2} \times 961 = 834 \times 480.5 = 400,617 \]

8. **Final Calculation**:
Now we need to subtract the two results:
\[ (60 \text{ therefore } 900) - (63 \text{ therefore } 898) = 403920 - 400617 = 3303 \]

### Conclusion:
The value of \( (60 \text{ therefore } 900) - (63 \text{ therefore } 898) \) is \( 3303 \).