What is the sum of all numbers in the sequence 3,6,9,12,15,...294,297,300

The given sequence is an **arithmetic sequence**, where the first term \(a_1 = 3\), the common difference \(d = 3\), and the last term \(a_n = 300\).

To find the **sum of the sequence**, we need two things:
1. The number of terms in the sequence (\(n\)).
2. The sum formula for an arithmetic sequence.

### Step 1: Find the number of terms (\(n\))

The general formula for the \(n\)-th term of an arithmetic sequence is:

\[
a_n = a_1 + (n - 1) \cdot d
\]

Substitute the known values (\(a_n = 300\), \(a_1 = 3\), \(d = 3\)):

\[
300 = 3 + (n - 1) \cdot 3
\]

Simplifying:

\[
300 = 3 + 3n - 3
\]

\[
300 = 3n
\]

\[
n = \frac{300}{3} = 100
\]

There are **100 terms** in the sequence.

### Step 2: Apply the sum formula

The sum \(S_n\) of an arithmetic sequence is given by:

\[
S_n = \frac{n}{2} \cdot (a_1 + a_n)
\]

Substitute the known values (\(n = 100\), \(a_1 = 3\), \(a_n = 300\)):

\[
S_{100} = \frac{100}{2} \cdot (3 + 300)
\]

\[
S_{100} = 50 \cdot 303 = 15,150
\]

### Final Answer:
The sum of all the numbers in the sequence is **15,150**.

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Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How do you find the common difference in an arithmetic sequence?
2. What would the sum be if the sequence started from a different number, say 5?
3. How does the sum formula change if the common difference is negative?
4. Can the sum formula be used for geometric sequences as well?
5. How can you check if a sequence is arithmetic?

**Tip:** Always double-check your calculations by substituting values back into the sequence formula.

Learn how to calculate the sum of the arithmetic sequence 3,6,9,...,300 using the sum formula for arithmetic sequences. This problem includes key concepts such as the n-th term formula and step-by-step calculations. Suitable for students in Grades 7-10.

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