A quadratic sequence is defined with the following properties: $$ T_{2}-T_{1}=7 $$ $$ T_{3}-T_{2}=13 $$ $$ T_{4}-T_{3}=19 $$

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To determine the general form of the quadratic sequence given the differences:

$$
\begin{align*}
T_2 - T_1 &= 7 \
T_3 - T_2 &= 13 \
T_4 - T_3 &= 19 \
\end{align*}
$$

Let's follow these steps:

### 1. Identify the Nature of the Sequence
A **quadratic sequence** has a constant **second difference**. Let's calculate the first and second differences:

- **First Differences:**
  $$
  d_1 = T_2 - T_1 = 7 \
  d_2 = T_3 - T_2 = 13 \
  d_3 = T_4 - T_3 = 19 \
  $$

- **Second Differences:**
  $$
  d_2 - d_1 = 13 - 7 = 6 \
  d_3 - d_2 = 19 - 13 = 6 \
  $$
  
The second difference is **constant and equal to 6**, confirming it's a quadratic sequence.

### 2. Determine the General Form
The general term of a quadratic sequence can be expressed as:

$$
T_n = an^2 + bn + c
$$

Given that the second difference is $$2a = 6$$, we find:

$$
a = 3
$$

So,

$$
T_n = 3n^2 + bn + c
$$

### 3. Find Coefficients $$b$$ and $$c$$
Using the known differences:

- For $$n = 1$$:
  $$
  T_1 = 3(1)^2 + b(1) + c = 3 + b + c
  $$
  
- For $$n = 2$$:
  $$
  T_2 = 3(2)^2 + b(2) + c = 12 + 2b + c
  $$
  
- For $$n = 3$$:
  $$
  T_3 = 3(3)^2 + b(3) + c = 27 + 3b + c
  $$

Given the differences:
$$
T_2 - T_1 = 7 \implies (12 + 2b + c) - (3 + b + c) = 9 + b = 7 \implies b = -2
$$
$$
T_3 - T_2 = 13 \implies (27 + 3b + c) - (12 + 2b + c) = 15 + b = 13 \implies b = -2
$$

With $$b = -2$$, the first term becomes:
$$
3 + (-2) + c = 1 + c = T_1
$$

Assuming $$T_1$$ is chosen such that $$c = 0$$ (for simplicity), we get:
$$
T_n = 3n^2 - 2n
$$

### 4. Verification
Let's verify the first few terms:

- $$T_1 = 3(1)^2 - 2(1) = 1$$
- $$T_2 = 3(2)^2 - 2(2) = 12 - 4 = 8$$
- $$T_3 = 3(3)^2 - 2(3) = 27 - 6 = 21$$
- $$T_4 = 3(4)^2 - 2(4) = 48 - 8 = 40$$

And the differences:
$$
T_2 - T_1 = 8 - 1 = 7 \
T_3 - T_2 = 21 - 8 = 13 \
T_4 - T_3 = 40 - 21 = 19 \
$$
These match the given differences.

### **Final Answer**
The $$n$$th term of the sequence is $$T_n = 3n² - 2n$$.
The $$n$$th term of the sequence is $$T_n = 3n² - 2n$$.

A quadratic sequence is defined with the following properties: \( T_{2}-T_{1}=7 \) \( T_{3}-T_{2}=13 \) \( T_{4}-T_{3}=19 \)

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