To find the sum of all the integers between 60 and 159 that are not multiples of 3, we can break this problem down into a step-by-step process.

Step 1: Identify the range of integers

We are interested in the integers between 60 and 159, inclusive. This means we are working with the range $[60, 159]$.

Step 2: Sum all the integers from 60 to 159

The formula for the sum of an arithmetic sequence is:

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

Where:

• $n$ is the number of terms,
• $a$ is the first term,
• $l$ is the last term.

The first term $a = 60$ and the last term $l = 159$.

The number of terms $n$ is: $n = 159 - 60 + 1 = 100$

Now, we can calculate the sum of all integers between 60 and 159:

$S = \frac{100}{2} \times (60 + 159) = 50 \times 219 = 10,950$

Step 3: Sum of multiples of 3 from 60 to 159

The multiples of 3 between 60 and 159 form an arithmetic sequence with first term 60 and common difference 3. To find the last multiple of 3 in this range, we divide 159 by 3 and round down:

$\left\lfloor \frac{159}{3} \right\rfloor = 53 \quad \text{so the last multiple of 3 is } 53 \times 3 = 159.$

Thus, the multiples of 3 from 60 to 159 are: $60, 63, 66, \ldots, 159$

This is an arithmetic sequence where $a = 60$, $d = 3$, and the last term is 159. The number of terms is:

$n = \frac{159 - 60}{3} + 1 = \frac{99}{3} + 1 = 34$

The sum of these multiples of 3 is:

$S = \frac{34}{2} \times (60 + 159) = 17 \times 219 = 3,723$

Step 4: Subtract the sum of multiples of 3 from the total sum

Now, we subtract the sum of the multiples of 3 from the sum of all integers in the range:

$\text{Sum of integers not multiples of 3} = 10,950 - 3,723 = 7,227$

Thus, the sum of all the integers between 60 and 159 that are not multiples of 3 is 7,227.

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Would you like further clarification on any of the steps or more details?

Here are five related questions to further your understanding:

1. How would you find the sum of integers between two arbitrary numbers?
2. What is the method to determine how many multiples of a number lie between two given values?
3. How can you generalize the approach to exclude other multiples besides 3?
4. What would be the sum of integers that are multiples of 5 between two numbers?
5. Can you find the sum of integers between 60 and 159 that are multiples of both 3 and 5?

Tip: Understanding the arithmetic sequence formula helps solve many problems related to sums and averages of evenly spaced numbers.