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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.

Work out the nth term of the given quadratic sequence: 7, 12, 19, 28, 39, ...

Learn how to find the nth term of the quadratic sequence 7, 12, 19, 28, 39 by using the method of finite differences. Follow the step-by-step guide to derive the general quadratic formula for this sequence. Suitable for students in Grades 8-10.

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