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The sequence provided is a quadratic sequence: 7, 10, 15, 22, 31, ...

### Step 1: Identify the first and second differences

Let's calculate the first and second differences:

- First differences:
  \( 10 - 7 = 3 \)
  \( 15 - 10 = 5 \)
  \( 22 - 15 = 7 \)
  \( 31 - 22 = 9 \)

- Second differences:
  \( 5 - 3 = 2 \)
  \( 7 - 5 = 2 \)
  \( 9 - 7 = 2 \)

Since the second differences are constant and equal to 2, we confirm that this is a quadratic sequence.

### Step 2: General form of a quadratic sequence

The general form of a quadratic sequence is:

\[
a_n = an^2 + bn + c
\]

Where \(a\), \(b\), and \(c\) are constants to be determined. We now have to solve for \(a\), \(b\), and \(c\) using the given terms of the sequence.

### Step 3: Set up the system of equations

Using the first three terms of the sequence:

- For \(n = 1\), \(a_1 = 7\):
  \[
  a(1)^2 + b(1) + c = 7 \quad \Rightarrow \quad a + b + c = 7
  \]

- For \(n = 2\), \(a_2 = 10\):
  \[
  a(2)^2 + b(2) + c = 10 \quad \Rightarrow \quad 4a + 2b + c = 10
  \]

- For \(n = 3\), \(a_3 = 15\):
  \[
  a(3)^2 + b(3) + c = 15 \quad \Rightarrow \quad 9a + 3b + c = 15
  \]

### Step 4: Solve the system of equations

We now solve the system of equations:

1. \( a + b + c = 7 \)
2. \( 4a + 2b + c = 10 \)
3. \( 9a + 3b + c = 15 \)

By solving these equations step-by-step, we will find the values of \(a\), \(b\), and \(c\).

Let's solve it.The solution to the system of equations is:

\[
a = 1, \quad b = 0, \quad c = 6
\]

Thus, the \(n\)th term rule for the quadratic sequence is:

\[
a_n = n^2 + 6
\]

### Step 5: Find the position of the term with value 150

Now, to find the position of the term with value 150, we set \(a_n = 150\):

\[
n^2 + 6 = 150
\]

Solving for \(n\):

\[
n^2 = 150 - 6 = 144
\]

\[
n = \sqrt{144} = 12
\]

Therefore, the term with value 150 is the 12th term in the sequence.

Would you like further clarification on any part of this solution?

Here are 5 related questions for further practice:
1. How would the formula change if the sequence started with a different number?
2. Can you determine the 15th term in this sequence using the formula?
3. What are the differences between a linear and quadratic sequence in terms of their nth term formulas?
4. How would you find the position of a different number, such as 200, in this sequence?
5. Can you derive the nth term formula for a cubic sequence?

**Tip**: For any quadratic sequence, the second differences between terms are always constant, which is the key to identifying and solving for the nth term.

The start of a quadratic sequence is shown below. By first finding the nth term rule, work out the position of the term with value 150 in the sequence: 7, 10, 15, 22, 31, ...

Discover how to determine the nth term of a quadratic sequence and find the position of the term with value 150. This problem breaks down quadratic sequences, system of equations, and applying algebraic methods. Suitable for students in Grades 9-11.

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