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To find the sum of all integers between 1 and 100 that are divisible by 7, let's analyze the sequence of these numbers and calculate their sum.

### Solution:

1. **Identify the integers divisible by 7 between 1 and 100:**
   - The smallest integer divisible by 7 in this range is 7.
   - The largest integer divisible by 7 in this range is 98.
   - So, we have the sequence: 7, 14, 21, ..., 98.

2. **Set up the sequence as an arithmetic series:**
   - This sequence is an arithmetic sequence where:
     - The first term \( a = 7 \).
     - The common difference \( d = 7 \).
     - The last term \( l = 98 \).

3. **Calculate the number of terms in the sequence:**
   - The formula for the \(n\)-th term of an arithmetic sequence is:
     \[
     a_n = a + (n - 1)d
     \]
   - Set \( a_n = 98 \) and solve for \( n \):
     \[
     98 = 7 + (n - 1) \cdot 7
     \]
     \[
     98 - 7 = (n - 1) \cdot 7
     \]
     \[
     91 = (n - 1) \cdot 7
     \]
     \[
     n - 1 = \frac{91}{7} = 13
     \]
     \[
     n = 14
     \]
   - So, there are 14 terms.

4. **Calculate the sum of the arithmetic sequence:**
   - The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic sequence is:
     \[
     S_n = \frac{n}{2} \cdot (a + l)
     \]
   - Substituting the values:
     \[
     S_{14} = \frac{14}{2} \cdot (7 + 98)
     \]
     \[
     S_{14} = 7 \cdot 105
     \]
     \[
     S_{14} = 735
     \]

### Answer:
The sum of all integers between 1 and 100 that are divisible by 7 is **735**.

Would you like a further breakdown of any step?

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Here are five related questions to explore:

1. What is the sum of all integers between 1 and 200 that are divisible by 7?
2. How would you find the sum of integers between 1 and 100 that are divisible by 5?
3. Can you calculate the average of all integers between 1 and 100 that are divisible by 7?
4. What is the product of the integers between 1 and 50 that are divisible by 7?
5. How would you find the sum of integers between 50 and 150 that are divisible by 9?

**Tip:** When dealing with sums of sequences, identifying the type (arithmetic, geometric) helps in choosing the right formula for efficient calculations.

The sum of all the integers between 1 and 100 which are divisible by 7

Calculate the sum of all integers between 1 and 100 that are divisible by 7. Learn step-by-step methods, including the use of arithmetic sequences, for students in Grades 7-9.

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