QUESTION 4 Given the quadratic number pattern: \( 69 ; 0-63 ; \ldots \) \( 4.1 \quad \) Write down the value of the next term in the pattern. \( 4.2 \quad \) Calculate an expression for the \( n^{\text {th }} \) term of the quadratic pattern. \( 4.3 \quad \) Determine the value of the SMALLEST term in this pattern.

Let's analyze the given quadratic number pattern step by step. ### **Given Pattern:** \( 69; \ 0; \ -63; \ \ldots \) We'll assume the sequence corresponds to \( n = 1, 2, 3, \ldots \) with the respective terms. ### **4.1 Write down the value of the next term in the pattern.** To find the next term, we'll first determine the quadratic expression that describes the pattern. #### **Finding the Quadratic Expression:** Assume the \( n^{\text{th}} \) term is given by: \[ a(n) = An^2 + Bn + C \] Using the given terms: 1. For \( n = 1 \): \[ A(1)^2 + B(1) + C = 69 \] \[ A + B + C = 69 \quad \text{(Equation 1)} \] 2. For \( n = 2 \): \[ A(2)^2 + B(2) + C = 0 \] \[ 4A + 2B + C = 0 \quad \text{(Equation 2)} \] 3. For \( n = 3 \): \[ A(3)^2 + B(3) + C = -63 \] \[ 9A + 3B + C = -63 \quad \text{(Equation 3)} \] **Solving the Equations:** - Subtract **Equation 1** from **Equation 2**: \[ (4A + 2B + C) - (A + B + C) = 0 - 69 \] \[ 3A + B = -69 \quad \text{(Equation 4)} \] - Subtract **Equation 2** from **Equation 3**: \[ (9A + 3B + C) - (4A + 2B + C) = -63 - 0 \] \[ 5A + B = -63 \quad \text{(Equation 5)} \] - Subtract **Equation 4** from **Equation 5**: \[ (5A + B) - (3A + B) = -63 - (-69) \] \[ 2A = 6 \] \[ A = 3 \] - Substitute \( A = 3 \) into **Equation 4**: \[ 3(3) + B = -69 \] \[ 9 + B = -69 \] \[ B = -78 \] - Substitute \( A = 3 \) and \( B = -78 \) into **Equation 1**: \[ 3 - 78 + C = 69 \] \[ C = 144 \] **Quadratic Expression:** \[ a(n) = 3n^2 - 78n + 144 \] #### **Finding the 4th Term (\( n = 4 \)):** \[ a(4) = 3(4)^2 - 78(4) + 144 \] \[ a(4) = 3(16) - 312 + 144 \] \[ a(4) = 48 - 312 + 144 \] \[ a(4) = -120 \] **Answer to 4.1:** \[ \boxed{-120} \] --- ### **4.2 Calculate an expression for the \( n^{\text{th}} \) term of the quadratic pattern.** From the previous analysis, the \( n^{\text{th}} \) term is: \[ a(n) = 3n^2 - 78n + 144 \] **Answer to 4.2:** \[ \boxed{a(n) = 3n^2 - 78n + 144} \] --- ### **4.3 Determine the value of the SMALLEST term in this pattern.** Since the quadratic coefficient \( A = 3 \) is positive, the parabola opens upwards, and the smallest term occurs at the vertex. #### **Finding the Vertex:** The vertex of a quadratic \( a(n) = An^2 + Bn + C \) occurs at: \[ n = -\frac{B}{2A} \] Substituting \( A = 3 \) and \( B = -78 \): \[ n = -\frac{-78}{2 \times 3} \] \[ n = \frac{78}{6} \] \[ n = 13 \] #### **Calculating \( a(13) \):** \[ a(13) = 3(13)^2 - 78(13) + 144 \] \[ a(13) = 3(169) - 1014 + 144 \] \[ a(13) = 507 - 1014 + 144 \] \[ a(13) = -363 \] **Answer to 4.3:** \[ \boxed{-363} \]