Use the formula to find the sum of the arithmetic series.

 

13+20+27+...+55

To find the sum of the arithmetic series \( 13 + 20 + 27 + \dots + 55 \), we can use the formula for the sum of an arithmetic series:

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

Where:
- \( S_n \) is the sum of the series,
- \( n \) is the number of terms,
- \( a_1 \) is the first term,
- \( a_n \) is the last term.

### Step 1: Identify the given values
- First term (\( a_1 \)) = 13
- Last term (\( a_n \)) = 55
- Common difference (\( d \)) = 7 (since 20 - 13 = 7)

### Step 2: Find the number of terms (\( n \))

We can use the formula for the \( n \)-th term of an arithmetic sequence:

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

Substitute the known values:

\[
55 = 13 + (n - 1) \times 7
\]

Solve for \( n \):

\[
55 - 13 = (n - 1) \times 7
\]
\[
42 = (n - 1) \times 7
\]
\[
n - 1 = 6
\]
\[
n = 7
\]

### Step 3: Calculate the sum using the formula

Now that we know \( n = 7 \), \( a_1 = 13 \), and \( a_n = 55 \), substitute these values into the sum formula:

\[
S_7 = \frac{7}{2} \times (13 + 55) = \frac{7}{2} \times 68 = 7 \times 34 = 238
\]

### Final Answer:
The sum of the arithmetic series is \( \boxed{238} \).

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Would you like a detailed explanation of any specific step, or do you have any other questions? 

Here are 5 related questions:
1. How do you determine the common difference of an arithmetic sequence?
2. How can you find the \( n \)-th term of an arithmetic sequence?
3. What is the formula for the sum of an infinite arithmetic series?
4. How does the sum of an arithmetic series differ from the sum of a geometric series?
5. Can the formula for the sum of an arithmetic series be applied to negative common differences?

**Tip**: Always check the values for the common difference and make sure the last term is part of the sequence before calculating the sum.

Use the formula to find the sum of the arithmetic series: 13+20+27+...+55

Learn how to calculate the sum of the arithmetic series 13+20+27+...+55 using key formulas and step-by-step reasoning. This problem highlights arithmetic series concepts and sequence properties, ideal for students in Grades 8-10.

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