Mathematics/Baseline QUESTION 2 Consider the quadratic number pattern: \( 5 ; 14 ; 27: 44 ; \ldots \) \begin{tabular}{l} 2.1 \\ \hline Decade 12\end{tabular} Determine the general term, \( T_{n} \), in the form \( T_{n}=a n^{2}+b n+0 \)

To determine the general term \( T_{n} \) of the quadratic number pattern \( 5, 14, 27, 44, \ldots \) in the form \( T_{n} = a n^{2} + b n \), follow these steps: 1. **Identify the Terms:** - For \( n = 1 \): \( T_1 = 5 \) - For \( n = 2 \): \( T_2 = 14 \) - For \( n = 3 \): \( T_3 = 27 \) - For \( n = 4 \): \( T_4 = 44 \) 2. **Set Up the Equations:** Using the general form \( T_{n} = a n^{2} + b n \), plug in the known values: - For \( n = 1 \): \( a(1)^2 + b(1) = 5 \) ⟹ \( a + b = 5 \)  (Equation 1) - For \( n = 2 \): \( a(2)^2 + b(2) = 14 \) ⟹ \( 4a + 2b = 14 \)  (Equation 2) 3. **Solve the System of Equations:** - From Equation 1: \( b = 5 - a \) - Substitute \( b \) into Equation 2: \[ 4a + 2(5 - a) = 14 \\ 4a + 10 - 2a = 14 \\ 2a = 4 \\ a = 2 \] - Substitute \( a = 2 \) back into Equation 1: \[ 2 + b = 5 \\ b = 3 \] 4. **Verify with Additional Terms:** - For \( n = 3 \): \[ T_3 = 2(3)^2 + 3(3) = 18 + 9 = 27 \] - For \( n = 4 \): \[ T_4 = 2(4)^2 + 3(4) = 32 + 12 = 44 \] Both terms match the given sequence, confirming our solution. **Therefore, the general term is:** \[ T_{n} = 2n^{2} + 3n \] **Answer:** After simplifying, the general term is Tₙ = 2 n² plus three times n. Thus, Tₙ = 2n² + 3n