etermine the genercl term for the following quadratic sequence using METHOD 1 \[ 5 ; 9 ; 17 ; 29 ; 45 \]

To determine the general term of the quadratic sequence \( 5, 9, 17, 29, 45 \) using METHOD 1, we will follow these steps: 1. **Identify the sequence**: The given sequence is \( a_1 = 5, a_2 = 9, a_3 = 17, a_4 = 29, a_5 = 45 \). 2. **Calculate the first differences**: We find the differences between consecutive terms. \[ \begin{align*} a_2 - a_1 & = 9 - 5 = 4 \\ a_3 - a_2 & = 17 - 9 = 8 \\ a_4 - a_3 & = 29 - 17 = 12 \\ a_5 - a_4 & = 45 - 29 = 16 \\ \end{align*} \] So, the first differences are \( 4, 8, 12, 16 \). 3. **Calculate the second differences**: Now, we find the differences of the first differences. \[ \begin{align*} 8 - 4 & = 4 \\ 12 - 8 & = 4 \\ 16 - 12 & = 4 \\ \end{align*} \] The second differences are constant and equal to \( 4 \). 4. **Formulate the general term**: Since the second differences are constant, the general term of the quadratic sequence can be expressed in the form: \[ a_n = An^2 + Bn + C \] where \( A \) is half of the second difference. Thus, \( A = \frac{4}{2} = 2 \). 5. **Set up equations to find \( B \) and \( C \)**: We can use the first few terms of the sequence to create a system of equations. We have: \[ \begin{align*} a_1 & = 2(1^2) + B(1) + C = 5 \quad \text{(1)} \\ a_2 & = 2(2^2) + B(2) + C = 9 \quad \text{(2)} \\ a_3 & = 2(3^2) + B(3) + C = 17 \quad \text{(3)} \\ \end{align*} \] Substituting the values into the equations: - From (1): \( 2 + B + C = 5 \) → \( B + C = 3 \) (Equation 1) - From (2): \( 8 + 2B + C = 9 \) → \( 2B + C = 1 \) (Equation 2) - From (3): \( 18 + 3B + C = 17 \) → \( 3B + C = -1 \) (Equation 3) 6. **Solve the system of equations**: We can solve Equations 1 and 2 first: \[ \begin{align*} B + C & = 3 \quad \text{(1)} \\ 2B + C & = 1 \quad \text{(2)} \\ \end{align*} \] Subtract Equation 1 from Equation 2: \[ (2B + C) - (B + C) = 1 - 3 \\ B = -2 \] Substitute \( B = -2 \) into Equation 1: \[ -2 + C = 3 \\ C = 5 \] 7. **Final general term**: Now we have \( A = 2 \), \( B = -2 \), and \( C = 5 \). Therefore, the general term of the sequence is: \[ a_n = 2n^2 - 2n + 5 \] Thus, the general term for the quadratic sequence \( 5, 9, 17, 29, 45 \) is: \[ \boxed{2n^2 - 2n + 5} \]