To find the first term (\(a\)) and the common difference (\(d\)) of an arithmetic sequence, we can use the formulas for the sum of the first \(n\) terms of the sequence.

The sum of the first \(n\) terms (\(S_n\)) of an arithmetic sequence is given by:

\[
S_n = \frac{n}{2} \left(2a + (n - 1)d\right)
\]

Given:
- \(S_{36} = 288\)
- \(S_{99} = -5445\)

Let's set up the equations based on the given values:

1. For \(S_{36}\):

\[
S_{36} = \frac{36}{2} \left(2a + (36 - 1)d\right) = 288
\]

Simplifying this equation:

\[
18 \left(2a + 35d\right) = 288
\]

\[
2a + 35d = \frac{288}{18}
\]

\[
2a + 35d = 16 \quad \text{(Equation 1)}
\]

2. For \(S_{99}\):

\[
S_{99} = \frac{99}{2} \left(2a + (99 - 1)d\right) = -5445
\]

Simplifying this equation:

\[
49.5 \left(2a + 98d\right) = -5445
\]

\[
2a + 98d = \frac{-5445}{49.5}
\]

\[
2a + 98d = -110 \quad \text{(Equation 2)}
\]

Now, we have two equations:

1. \(2a + 35d = 16\)
2. \(2a + 98d = -110\)

Let's solve this system of equations by eliminating \(2a\):

Subtract Equation 1 from Equation 2:

\[
(2a + 98d) - (2a + 35d) = -110 - 16
\]

\[
98d - 35d = -126
\]

\[
63d = -126
\]

\[
d = \frac{-126}{63}
\]

\[
d = -2
\]

Now, substitute \(d = -2\) into Equation 1 to find \(a\):

\[
2a + 35(-2) = 16
\]

\[
2a - 70 = 16
\]

\[
2a = 16 + 70
\]

\[
2a = 86
\]

\[
a = \frac{86}{2}
\]

\[
a = 43
\]

So, the first term \(a\) is 43, and the common difference \(d\) is -2.

### Summary
- \(a = 43\)
- \(d = -2\)

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What would be the 36th term of this arithmetic sequence?
2. How can we find the sum of the first 50 terms of this sequence?
3. If the 100th term of this sequence is needed, how can we calculate it?
4. What is the general formula for the \(n\)th term of an arithmetic sequence?
5. How does changing the common difference \(d\) affect the sum of an arithmetic sequence?

**Tip:** In arithmetic sequences, the common difference \(d\) determines whether the sequence increases or decreases and by how much.

Learn how to find the first term (a) and common difference (d) of an arithmetic sequence using the sum formulas given S36 = 288 and S99 = -5445. This problem involves solving equations derived from arithmetic sequence properties. Suitable for high school students studying sequences and series.

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