Which of the following is the sum of all the multiples of 3 from 15 to 48?

To find the sum of all the multiples of 3 from 15 to 48, follow these steps:

### Step 1: Identify the multiples of 3 in the range 15 to 48
The first multiple of 3 in this range is 15, and the last multiple of 3 in this range is 48.

To list the multiples:
\[
15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48
\]

### Step 2: Determine the number of terms
This is an arithmetic sequence where the first term \(a_1\) is 15, the last term \(a_n\) is 48, and the common difference \(d\) is 3.

To find the number of terms \(n\), use the formula for the nth term of an arithmetic sequence:
\[
a_n = a_1 + (n-1)d
\]
Substituting the values:
\[
48 = 15 + (n-1) \times 3
\]
\[
48 - 15 = (n-1) \times 3
\]
\[
33 = (n-1) \times 3
\]
\[
n-1 = \frac{33}{3} = 11
\]
\[
n = 11 + 1 = 12
\]

So, there are 12 terms.

### Step 3: Calculate the sum of the sequence
The sum \(S_n\) of an arithmetic sequence is given by:
\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]
Substituting the values:
\[
S_{12} = \frac{12}{2} \times (15 + 48)
\]
\[
S_{12} = 6 \times 63 = 378
\]

Thus, the sum of all the multiples of 3 from 15 to 48 is **378**.

Do you want more details or have any other questions?

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2. What is the sum of all the multiples of 5 between 10 and 100?
3. How can you find the sum of an arithmetic series if you only know the first term and the common difference?
4. What is the sum of all even numbers from 10 to 50?
5. How do you determine if a number is a multiple of 3?
6. What is the sum of the first 50 positive integers?
7. How does the formula for the sum of an arithmetic series differ from that of a geometric series?
8. What is the formula for the sum of all multiples of a given number in a specified range?

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

Learn how to calculate the sum of all multiples of 3 from 15 to 48 using arithmetic sequence principles and formulas. This problem involves identifying multiples of 3, determining the number of terms in the sequence, and applying the formula for the sum of an arithmetic sequence. Suitable for Grades 7-9 students.

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