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?

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.