11 The sum to $$ n $$ terms of an arithmetic sequence is given by: $$ S_{n}=63 n-3 n^{2} $$ a Calculate the 4th term. b Determine the value(s) of $$ n $$ if the sum of $$ n $$ terms is -300 .

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Let's solve the problem step-by-step.

**Given:**
The sum of the first $$ n $$ terms of an arithmetic sequence is:
$$
S_{n} = 63n - 3n^{2}
$$

### Part a: Calculate the 4th Term ($$ t_4 $$)

**Step 1: Understand the Relationship**

For an arithmetic sequence:
$$
S_{n} = \frac{n}{2} [2a + (n-1)d]
$$
where:
- $$ a $$ is the first term.
- $$ d $$ is the common difference.
- $$ t_n = a + (n-1)d $$ is the $$ n^{th} $$ term.

**Step 2: Equate the Given Sum to the Arithmetic Sum Formula**
$$
\frac{n}{2} [2a + (n-1)d] = 63n - 3n^{2}
$$

**Step 3: Simplify the Equation**
Divide both sides by $$ n $$ (assuming $$ n \neq 0 $$):
$$
\frac{1}{2} [2a + (n-1)d] = 63 - 3n
$$
Multiply both sides by 2:
$$
2a + (n - 1)d = 126 - 6n
$$
$$
2a + dn - d = 126 - 6n
$$

**Step 4: Equate Coefficients**

For the equation to hold for all $$ n $$, the coefficients of like terms must be equal:
- Coefficient of $$ n $$:
  $$
  d = -6
  $$
- Constant term:
  $$
  2a - d = 126
  $$
  Substitute $$ d = -6 $$:
  $$
  2a - (-6) = 126 \
  2a + 6 = 126 \
  2a = 120 \
  a = 60
  $$

**Step 5: Find the 4th Term**
$$
t_4 = a + (4-1)d = 60 + 3(-6) = 60 - 18 = 42
$$

**Answer to Part a:**  
The 4th term is **42**.

---

### Part b: Determine the Value(s) of $$ n $$ if $$ S_n = -300 $$

**Step 1: Set Up the Equation**
$$
63n - 3n^{2} = -300
$$
$$
-3n^{2} + 63n + 300 = 0
$$

**Step 2: Simplify the Equation**
Divide the entire equation by -3 to simplify:
$$
n^{2} - 21n - 100 = 0
$$

**Step 3: Solve the Quadratic Equation**
Use the quadratic formula:
$$
n = \frac{21 \pm \sqrt{(-21)^2 - 4 \cdot 1 \cdot (-100)}}{2 \cdot 1} \
n = \frac{21 \pm \sqrt{441 + 400}}{2} \
n = \frac{21 \pm \sqrt{841}}{2} \
n = \frac{21 \pm 29}{2}
$$

This gives two solutions:
1. $$ n = \frac{21 + 29}{2} = \frac{50}{2} = 25 $$
2. $$ n = \frac{21 - 29}{2} = \frac{-8}{2} = -4 $$

**Step 4: Determine Validity**

Since $$ n $$ represents the number of terms, it must be a positive integer.

**Answer to Part b:**  
The valid value of $$ n $$ is **25**.
**Part a:** The 4th term is **42**. **Part b:** The value of $$ n $$ is **25**.

11 The sum to \( n \) terms of an arithmetic sequence is given by: \( S_{n}=63 n-3 n^{2} \) a Calculate the 4th term. b Determine the value(s) of \( n \) if the sum of \( n \) terms is -300 .

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