The given series is:

\[
48 + 45 + 42 + \dots + (-81) + (-84)
\]

This is an arithmetic series because the difference between consecutive terms is constant. Let's break it down:

1. **First term (a)**: The first term is \( 48 \).
2. **Common difference (d)**: The common difference is \( 45 - 48 = -3 \).
3. **Last term (l)**: The last term is \( -84 \).

### Step 1: Find the number of terms (n)
We use the formula for the \(n\)-th term of an arithmetic sequence:

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

Substitute the known values:

\[
-84 = 48 + (n-1)(-3)
\]

Simplify:

\[
-84 = 48 - 3(n-1)
\]

\[
-84 = 48 - 3n + 3
\]

\[
-84 = 51 - 3n
\]

\[
-84 - 51 = -3n
\]

\[
-135 = -3n
\]

\[
n = 45
\]

So, there are 45 terms in the series.

### Step 2: Sum of the arithmetic series
The sum of an arithmetic series is given by the formula:

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

Substitute the known values:

\[
S_{45} = \frac{45}{2} \times (48 + (-84))
\]

\[
S_{45} = \frac{45}{2} \times (-36)
\]

\[
S_{45} = 45 \times (-18)
\]

\[
S_{45} = -810
\]

### Final Answer:
The sum of the series is \( \boxed{-810} \).

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What if the common difference in the arithmetic series was positive?
2. How do you find the sum of a geometric series?
3. Can you find the sum of the first 50 terms of a similar arithmetic sequence?
4. How would the sum change if the last term was different?
5. What is the difference between an arithmetic and a geometric progression?

**Tip**: In arithmetic series, knowing the number of terms is crucial to calculating the sum effectively!

Explore how to find the sum of the arithmetic series 48 + 45 + 42 + ... + (-84). This problem highlights key mathematical concepts such as arithmetic sequences, the nth-term formula, and the summation formula. Perfect for students in Grades 9-11.

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