Find the sum of all the integers between 60 and 159 that are not multiples of three.

To find the sum of all integers between 60 and 159 that are not multiples of three, we can follow these steps:

### Step 1: Define the Range

The range of integers is from 60 to 159. This can be represented as:

\[
\text{Range} = \{60, 61, 62, \ldots, 159\}
\]

### Step 2: Calculate the Total Sum of All Integers in the Range

We use the formula for the sum of an arithmetic sequence:

\[
S = \frac{n}{2} \times (a + l)
\]

where:
- \( n \) is the number of terms
- \( a \) is the first term (60)
- \( l \) is the last term (159)

First, calculate \( n \):

\[
n = 159 - 60 + 1 = 100
\]

Now, compute the total sum:

\[
S = \frac{100}{2} \times (60 + 159) = 50 \times 219 = 10950
\]

### Step 3: Find the Sum of Multiples of 3 in the Range

To find the sum of multiples of 3 between 60 and 159, identify the smallest and largest multiples of 3 within the range:

- Smallest multiple: 60
- Largest multiple: 159

These multiples form an arithmetic sequence with the first term \( a = 60 \) and the common difference \( d = 3 \). To find the number of terms \( n \):

\[
n = \frac{159 - 60}{3} + 1 = 34
\]

Now, calculate the sum of these multiples:

\[
S = \frac{34}{2} \times (60 + 159) = 17 \times 219 = 3723
\]

### Step 4: Subtract the Sum of Multiples of 3 from the Total Sum

Finally, subtract the sum of multiples of 3 from the total sum to get the sum of integers that are not multiples of 3:

\[
\text{Sum} = 10950 - 3723 = 7227
\]

### Final Answer

The sum of all integers between 60 and 159 that are not multiples of three is:

\[
\boxed{7227}
\]

Would you like more details on any step, or do you have any questions?

### Related Questions

1. How would you find the sum of integers between two given numbers that are multiples of 5?
2. What is the sum of all even numbers between 50 and 200?
3. How do you calculate the sum of an arithmetic series with missing terms?
4. What is the sum of all prime numbers between 1 and 100?
5. How would you find the sum of all odd numbers between two given numbers?

### Tip

When working with arithmetic sequences, always identify the first term, last term, and the number of terms to use the sum formula efficiently!

Explore how to calculate the sum of all integers between 60 and 159 that are not multiples of three. This problem includes concepts from arithmetic sequences and number theory, using the formula for the sum of an arithmetic sequence. Suitable for students in Grades 9-11.

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