The question in the image asks to "Work out the $n$th term of the given quadratic sequence: 7, 12, 19, 28, 39, ...".

To solve this:

1. Identify the first few terms of the sequence: The sequence is 7, 12, 19, 28, 39.

2. Find the first differences: $12 - 7 = 5,\quad 19 - 12 = 7,\quad 28 - 19 = 9,\quad 39 - 28 = 11$ So, the first differences are 5, 7, 9, 11.

3. Find the second differences: $7 - 5 = 2,\quad 9 - 7 = 2,\quad 11 - 9 = 2$ The second differences are constant and equal to 2, confirming that this is a quadratic sequence.

4. Use the general form of a quadratic sequence: The $n$th term of a quadratic sequence is generally given by: $T_n = an^2 + bn + c$ where $a$, $b$, and $c$ are constants.

5. Solve for $a$, $b$, and $c$: Using the second difference rule for quadratic sequences, the value of $a$ is half of the second difference: $a = \frac{2}{2} = 1$ So, the general form becomes $T_n = n^2 + bn + c$.

6. Substitute known terms to solve for $b$ and $c$: Using the first three terms of the sequence:

   For $n = 1$, $T_1 = 7$: $1^2 + b(1) + c = 7 \quad \Rightarrow \quad 1 + b + c = 7 \quad \Rightarrow \quad b + c = 6 \quad (1)$

   For $n = 2$, $T_2 = 12$: $2^2 + b(2) + c = 12 \quad \Rightarrow \quad 4 + 2b + c = 12 \quad \Rightarrow \quad 2b + c = 8 \quad (2)$

   Subtract equation (1) from equation (2): $(2b + c) - (b + c) = 8 - 6 \quad \Rightarrow \quad b = 2$

   Substitute $b = 2$ into equation (1): $2 + c = 6 \quad \Rightarrow \quad c = 4$

7. Final formula: The formula for the $n$th term of the sequence is: $T_n = n^2 + 2n + 4$

Would you like further explanation on this? Here are some related questions:

1. How can we verify the formula for different values of $n$?
2. What is the sum of the first $n$ terms of a quadratic sequence?
3. How do first and second differences help in identifying the type of sequence?
4. What changes when we solve for the $n$th term of a cubic sequence?
5. Can we apply this method to any quadratic sequence?

Tip: For quadratic sequences, always check second differences to confirm its nature.