https://file.aiquickdraw.com/mathbot/file/1727345489173idcahp1w.jpg

The image shows a sum of multiples of 3, which forms an arithmetic series. It begins with:

\[
3 + 6 + 9 + \dots + 180
\]

This can be interpreted as an arithmetic sequence where:

- The first term \(a = 3\)
- The common difference \(d = 3\)
- The last term \(l = 180\)

We can use the formula for the sum \(S_n\) of an arithmetic series:

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

But first, we need to find \(n\), the number of terms. We can use the formula for the \(n\)-th term of an arithmetic sequence:

\[
l = a + (n-1) \times d
\]

Substituting the values:

\[
180 = 3 + (n-1) \times 3
\]

Now, let's solve for \(n\).The number of terms \(n = 60\). Now, we can find the sum of the series using the formula for the sum of an arithmetic sequence:

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

Substituting the values:

\[
S_{60} = \frac{60}{2} \times (3 + 180)
\]

Let's calculate the sum.The sum of the arithmetic series is \( S_{60} = 5490 \).

Would you like further details on this solution or have any other questions?

Here are five related questions to explore:
1. How do you find the sum of an arithmetic series with a negative common difference?
2. What changes if the sequence starts with a number other than 3?
3. Can you derive the formula for the sum of an arithmetic series?
4. How would you approach this problem if only partial information (like missing terms) was given?
5. What is the significance of arithmetic sequences in real-world applications?

**Tip:** In any arithmetic sequence, knowing the common difference and the first term allows you to determine any term or sum easily!

What is the sum of the arithmetic sequence 3 + 6 + 9 + ... + 180?

Learn how to calculate the sum of the arithmetic sequence 3 + 6 + 9 + ... + 180. This problem uses key formulas for arithmetic series and is suitable for students in Grades 8-10.

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