find the sum of all integers multiples of 7 from 60 to 290

To find the sum of all integers that are multiples of 7 between 60 and 290, follow these steps:

### 1. Identify the first and last multiples of 7 in the range:
- **First multiple of 7**: As we previously found, the smallest multiple of 7 greater than or equal to 60 is 63.
- **Last multiple of 7**: The largest multiple of 7 less than or equal to 290 is 287.

### 2. List the multiples of 7 in the range:
- The multiples of 7 are: 63, 70, 77, ..., 287.

### 3. Count the number of multiples of 7:
- The first multiple of 7 in the sequence is \( 63 = 7 \times 9 \).
- The last multiple of 7 in the sequence is \( 287 = 7 \times 41 \).
- The number of terms \( n \) is \( 41 - 9 + 1 = 33 \).

### 4. Use the formula for the sum of an arithmetic sequence:
The sum \( S \) of an arithmetic sequence is given by:
\[
S = \frac{n}{2} \times (\text{first term} + \text{last term})
\]
- Here, \( n = 33 \), the first term is 63, and the last term is 287.

### 5. Calculate the sum:
\[
S = \frac{33}{2} \times (63 + 287) = \frac{33}{2} \times 350 = 33 \times 175 = 5775
\]

So, the sum of all integers that are multiples of 7 between 60 and 290 is **5,775**.

Would you like any more details or have any questions?

Here are some related questions you might find interesting:
1. What is the sum of all multiples of 5 between 50 and 200?
2. How many integers between 1 and 1000 are multiples of both 7 and 11?
3. Find the sum of all multiples of 7 between 1 and 1000.
4. What is the smallest positive integer that is both a multiple of 7 and 9?
5. What is the product of all multiples of 7 between 63 and 287?

**Tip:** The sum of an arithmetic series can be quickly calculated if you know the first term, last term, and the number of terms.

Discover how to calculate the sum of all integers that are multiples of 7 between 60 and 290. Follow a detailed step-by-step solution including identifying multiples of 7, applying the sum formula for arithmetic sequences, and calculating the final sum. Suitable for advanced high school students.

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