To find the sum of all numbers between 500 and 1000 which are divisible by 13, we can follow these steps:
1. Identify the first number greater than or equal to 500 that is divisible by 13:
- The smallest integer that is greater than or equal to 500 and also divisible by 13 is 507.
2. Identify the last number less than or equal to 1000 that is divisible by 13:
- The largest integer that is less than or equal to 1000 and divisible by 13 is 988.
3. Determine the sequence of numbers divisible by 13 between 507 and 988:
- The sequence would go: 507, 520, 533, ..., 988.
- This is an arithmetic sequence where the first term [tex]\( a = 507 \)[/tex] and the common difference [tex]\( d = 13 \)[/tex].
4. Calculate the number of terms in this sequence:
- We can use the formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence: [tex]\( a_n = a + (n-1) \cdot d \)[/tex].
- Solving for [tex]\( n \)[/tex] when [tex]\( a_n = 988 \)[/tex]:
[tex]\[
988 = 507 + (n-1) \cdot 13 \\
988 - 507 = (n-1) \cdot 13 \\
481 = (n-1) \cdot 13 \\
n-1 = \frac{481}{13} \\
n-1 = 37 \\
n = 38
\][/tex]
5. Apply the formula for the sum of an arithmetic series:
- The sum of the first [tex]\( n \)[/tex] terms of an arithmetic sequence can be found using the formula:
[tex]\[
S_n = \frac{n}{2} \cdot (a + l)
\][/tex]
where [tex]\( n \)[/tex] is the number of terms, [tex]\( a \)[/tex] is the first term, and [tex]\( l \)[/tex] is the last term.
- Substituting the values:
[tex]\[
S_{38} = \frac{38}{2} \cdot (507 + 988) \\
S_{38} = 19 \cdot 1495 \\
S_{38} = 28405
\][/tex]
Therefore, the sum of all numbers between 500 and 1000 which are divisible by 13 is 28,405.
