Given the quadratic number pattern: $$ -5 ; -4 ; -1 ; 4 ; \ldots $$ 311 Determine the $$ n^{*} $$ term of the quadratic number patterns in the form $$ T_{n}=a n^{2}+b n+c $$.

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To determine the $$ n^{*} $$ term of the quadratic number pattern given by the sequence $$ -5, -4, -1, 4, \ldots $$, we will follow these steps:

1. **Identify the terms of the sequence**: 
   - $$ T_1 = -5 $$
   - $$ T_2 = -4 $$
   - $$ T_3 = -1 $$
   - $$ T_4 = 4 $$

2. **Set up the general form of the quadratic sequence**: 
   The general form of a quadratic sequence is given by:
   $$
   T_n = a n^2 + b n + c
   $$
   where $$ a $$, $$ b $$, and $$ c $$ are constants we need to determine.

3. **Create a system of equations**: 
   We can substitute the known terms into the quadratic formula to create a system of equations:
   - For $$ n = 1 $$:
     $$
     a(1^2) + b(1) + c = -5 \quad \Rightarrow \quad a + b + c = -5 \quad \text{(1)}
     $$
   - For $$ n = 2 $$:
     $$
     a(2^2) + b(2) + c = -4 \quad \Rightarrow \quad 4a + 2b + c = -4 \quad \text{(2)}
     $$
   - For $$ n = 3 $$:
     $$
     a(3^2) + b(3) + c = -1 \quad \Rightarrow \quad 9a + 3b + c = -1 \quad \text{(3)}
     $$

4. **Solve the system of equations**: 
   We will solve equations (1), (2), and (3) simultaneously.

From equation (1):
   $$
   c = -5 - a - b \quad \text{(4)}
   $$

Substitute (4) into equations (2) and (3):
   - Substituting into (2):
     $$
     4a + 2b + (-5 - a - b) = -4
     $$
     Simplifying:
     $$
     4a + 2b - 5 - a - b = -4 \quad \Rightarrow \quad 3a + b - 5 = -4 \quad \Rightarrow \quad 3a + b = 1 \quad \text{(5)}
     $$

- Substituting into (3):
     $$
     9a + 3b + (-5 - a - b) = -1
     $$
     Simplifying:
     $$
     9a + 3b - 5 - a - b = -1 \quad \Rightarrow \quad 8a + 2b - 5 = -1 \quad \Rightarrow \quad 8a + 2b = 4 \quad \Rightarrow \quad 4a + b = 2 \quad \text{(6)}
     $$

5. **Solve equations (5) and (6)**:
   Now we have a simpler system:
   - $$ 3a + b = 1 $$ (5)
   - $$ 4a + b = 2 $$ (6)

Subtract (5) from (6):
   $$
   (4a + b) - (3a + b) = 2 - 1 \quad \Rightarrow \quad a = 1
   $$

Substitute $$ a = 1 $$ back into (5):
   $$
   3(1) + b = 1 \quad \Rightarrow \quad 3 + b = 1 \quad \Rightarrow \quad b = -2
   $$

Substitute $$ a = 1 $$ and $$ b = -2 $$ into (4):
   $$
   c = -5 - 1 + 2 = -4
   $$

6. **Final formula**:
   We have determined:
   $$
   a = 1, \quad b = -2, \quad c = -4
   $$
   Thus, the formula for the $$ n^{*} $$ term is:
   $$
   T_n = n^2 - 2n - 4
   $$

Now, we can summarize the result:
The $$ n^{*} $$ term of the quadratic number pattern is given by:
$$
T_n = n^2 - 2n - 4
$$
The $$ n^{*} $$ term of the quadratic number pattern is $$ T_n = n^2 - 2n - 4 $$.

Given the quadratic number pattern: \( -5 ; -4 ; -1 ; 4 ; \ldots \) 311 Determine the \( n^{*} \) term of the quadratic number patterns in the form \( T_{n}=a n^{2}+b n+c \).

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