Find the sum of the multiplies of 100 from 200 to 400

To find the sum of the multiples of 100 between 200 and 400 (inclusive), we first identify these multiples:

- \( 200, 300, 400 \)

These numbers form an arithmetic sequence where:
- First term (\(a_1\)) = 200
- Common difference (\(d\)) = 100
- Last term (\(a_n\)) = 400

The number of terms (\(n\)) in the sequence can be found using the formula for the \(n\)-th term of an arithmetic sequence:

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

Substituting the known values:

\[
400 = 200 + (n-1) \cdot 100
\]

\[
400 - 200 = (n-1) \cdot 100
\]

\[
200 = (n-1) \cdot 100
\]

\[
n-1 = 2 \quad \Rightarrow \quad n = 3
\]

Thus, there are 3 terms. Now, the sum \(S_n\) of the terms of an arithmetic sequence can be calculated using the formula:

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

Substituting the values:

\[
S_3 = \frac{3}{2} \cdot (200 + 400) = \frac{3}{2} \cdot 600 = 3 \cdot 300 = 900
\]

Thus, the sum of the multiples of 100 from 200 to 400 is **900**.

Would you like further details or have any questions?

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Here are five related questions:
1. What is the sum of the multiples of 50 from 200 to 400?
2. How many multiples of 200 exist between 200 and 800?
3. What is the sum of all even numbers from 100 to 300?
4. Find the sum of the multiples of 150 between 150 and 450.
5. What is the sum of all multiples of 100 from 100 to 1000?

**Tip**: When working with arithmetic sequences, always check the first and last terms carefully to ensure you capture the full range.

Find the sum of the multiples of 100 from 200 to 400

Learn how to find the sum of multiples of 100 between 200 and 400 using arithmetic sequences. This detailed solution walks through the core concepts of sequence summation, providing key formulas and a step-by-step approach. Ideal for Grades 6-8.

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