Find the sum of this arithmetic series 147+143+139+...... + (-9)

To find the sum of the arithmetic series \(147 + 143 + 139 + \dots + (-9)\), we can use the following steps:

### Step 1: Identify the components of the arithmetic sequence
- **First term (\(a\))**: \(147\)
- **Common difference (\(d\))**: \(143 - 147 = -4\)
- **Last term (\(l\))**: \(-9\)

### Step 2: Determine the number of terms (\(n\))
The formula for the \(n\)th term of an arithmetic sequence is:

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

Given that the last term \(a_n = -9\), we can substitute the known values:

\[
-9 = 147 + (n-1) \cdot (-4)
\]

Now, solve for \(n\):

\[
-9 - 147 = (n-1) \cdot (-4)
\]

\[
-156 = (n-1) \cdot (-4)
\]

\[
n-1 = \frac{-156}{-4}
\]

\[
n-1 = 39
\]

\[
n = 40
\]

So, there are \(40\) terms in the series.

### Step 3: Calculate the sum of the arithmetic series
The sum \(S_n\) of an arithmetic series is given by:

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

Substitute the known values:

\[
S_{40} = \frac{40}{2} \cdot (147 + (-9))
\]

\[
S_{40} = 20 \cdot (147 - 9)
\]

\[
S_{40} = 20 \cdot 138
\]

\[
S_{40} = 2760
\]

### Final Answer:
The sum of the arithmetic series is **2760**.

Would you like further details or have any questions? 

Here are five related questions you might find interesting:
1. What is the sum of the first 50 terms of the series \(147 + 143 + 139 + \dots\)?
2. How do you find the sum if the last term is unknown?
3. What if the common difference were positive, how would the sum change?
4. How can you find the average of all terms in an arithmetic series?
5. How would you approach finding the sum if the sequence was geometric instead?

**Tip:** The sum of an arithmetic series can be quickly calculated if you know the first and last terms along with the number of terms.

Learn how to calculate the sum of the arithmetic series 147 + 143 + 139 + ... + (-9) step-by-step. This problem involves identifying the first term, common difference, last term, and using the formula for the sum of an arithmetic series. Suitable for high school students.

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