QUESTION 4 Consider the quadratic pattern \( -9 ;-6 ; 1 ; 12 ; x \) 4.1 Determine the value of \( x \). 4.2 Determine a formula for the \( n^{\text {th }} \) term of the pattern. 4.3 Which term in the sequence has a value of 244 ? 4.4 A new pattern \( P_{n} \) is formed by adding 3 to each term in the quadratic pattern. Write down the general term \( P_{n} \) in the form \( P_{n}=a n^{2}+b n+c \).

To solve the problem, we will analyze the quadratic pattern given: \(-9, -6, 1, 12, x\). ### 4.1 Determine the value of \( x \). To find \( x \), we first need to identify the pattern in the sequence. We can calculate the first differences and second differences. **First differences:** - \(-6 - (-9) = 3\) - \(1 - (-6) = 7\) - \(12 - 1 = 11\) - \(x - 12\) So, the first differences are: \(3, 7, 11, (x - 12)\). **Second differences:** - \(7 - 3 = 4\) - \(11 - 7 = 4\) - \((x - 12) - 11\) The second differences are constant at \(4\), which indicates that the sequence is quadratic. Since the second differences are constant, we can find \(x\) by setting the last second difference equal to \(4\): \[ (x - 12) - 11 = 4 \] \[ x - 12 - 11 = 4 \] \[ x - 23 = 4 \] \[ x = 27 \] Thus, the value of \( x \) is \( \boxed{27} \). ### 4.2 Determine a formula for the \( n^{\text{th}} \) term of the pattern. Since we have established that the sequence is quadratic, we can express the \( n^{\text{th}} \) term as: \[ T_n = an^2 + bn + c \] We can use the first few terms to set up a system of equations. We know: - \( T_1 = -9 \) - \( T_2 = -6 \) - \( T_3 = 1 \) This gives us the following equations: 1. \( a(1^2) + b(1) + c = -9 \) → \( a + b + c = -9 \) (1) 2. \( a(2^2) + b(2) + c = -6 \) → \( 4a + 2b + c = -6 \) (2) 3. \( a(3^2) + b(3) + c = 1 \) → \( 9a + 3b + c = 1 \) (3) Now we can solve this system of equations. Subtract (1) from (2): \[ (4a + 2b + c) - (a + b + c) = -6 + 9 \] \[ 3a + b = 3 \quad \text{(4)} \] Subtract (2) from (3): \[ (9a + 3b + c) - (4a + 2b + c) = 1 + 6 \] \[ 5a + b = 7 \quad \text{(5)} \] Now we can subtract (4) from (5): \[ (5a + b) - (3a + b) = 7 - 3 \] \[ 2a = 4 \quad \Rightarrow \quad a = 2 \] Substituting \( a = 2 \) into (4): \[ 3(2) + b = 3 \] \[ 6 + b = 3 \quad \Rightarrow \quad b = -3 \] Now substituting \( a = 2 \) and \( b = -3 \) into (1): \[ 2 - 3 + c = -9 \] \[ -1 + c = -9 \quad \Rightarrow \quad c = -8 \] Thus, the formula for the \( n^{\text{th}} \) term is: \[ T_n = 2n^2 - 3n - 8 \] ### 4.3 Which term in the sequence has a value of 244? We need to solve the equation: \[ 2n^2 - 3n - 8 = 244 \] Rearranging gives: \[ 2n^2 - 3n - 252 = 0 \] Now we can use the quadratic formula: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2, b = -3, c = -252 \): \[ n = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-252)}}{2(2)} \] \[ n = \frac{3 \pm \sqrt{9 + 2016}}{4} \] \[ n = \frac{3 \pm \sqrt{2025}}{4} \] \[ n = \frac{3 \pm 45}{4} \] Calculating the two possible values: 1. \( n = \frac{48}{4} = 12 \) 2. \( n = \frac{-42}{4} = -10.5 \) (not valid) Thus, the term in the sequence that has a value of 244 is the \( 12^{\text{th}} \) term. ### 4.4 A new pattern \( P_n \) is formed by adding 3 to each term in the quadratic pattern. The new pattern is: \[ P_n = T_n + 3 = (2n^2 - 3n - 8) + 3 \] \[ P_n = 2n^2 - 3n - 5 \] Thus, the general term \( P_n \) in the form \( P_n = an^2 + bn + c \) is: \[ P_n = 2n^2 - 3n - 5 \] In summary: - \( x = 27 \) - \( T_n = 2n^2 - 3n - 8 \) - The term with value 244 is the \( 12^{\text{th}} \) term. - \( P_n = 2n^2 - 3n - 5 \)