\( -1, \quad 2, \quad 7, \quad 14, \quad 23, \quad \ldots \) a) Work out the \( n^{\text {th }} \) term rule for this sequence. b) At what position does the term with value 62 appear in this sequence?

Let's analyze the given sequence step by step to address both parts of the problem. **Given Sequence:** \[ -1, \quad 2, \quad 7, \quad 14, \quad 23, \quad \ldots \] ### a) Finding the \( n^{\text{th}} \) Term Rule **Step 1: Determine the Pattern of Differences** First, let's find the differences between consecutive terms: \[ \begin{align*} 2 - (-1) &= 3 \\ 7 - 2 &= 5 \\ 14 - 7 &= 7 \\ 23 - 14 &= 9 \\ \end{align*} \] The differences are: \(3, 5, 7, 9, \ldots\). **Step 2: Identify the Nature of the Sequence** Since the differences between terms are increasing by a constant value (in this case, 2), the sequence is **quadratic**. This implies that the \( n^{\text{th}} \) term can be expressed in the form: \[ a_n = an^2 + bn + c \] **Step 3: Determine the Coefficients \( a \), \( b \), and \( c \)** We can set up equations using the known terms: 1. For \( n = 1 \): \[ a(1)^2 + b(1) + c = -1 \quad \Rightarrow \quad a + b + c = -1 \quad \text{(Equation 1)} \] 2. For \( n = 2 \): \[ a(2)^2 + b(2) + c = 2 \quad \Rightarrow \quad 4a + 2b + c = 2 \quad \text{(Equation 2)} \] 3. For \( n = 3 \): \[ a(3)^2 + b(3) + c = 7 \quad \Rightarrow \quad 9a + 3b + c = 7 \quad \text{(Equation 3)} \] **Solving the System of Equations:** Subtract Equation 1 from Equation 2: \[ (4a + 2b + c) - (a + b + c) = 2 - (-1) \quad \Rightarrow \quad 3a + b = 3 \quad \text{(Equation 4)} \] Subtract Equation 2 from Equation 3: \[ (9a + 3b + c) - (4a + 2b + c) = 7 - 2 \quad \Rightarrow \quad 5a + b = 5 \quad \text{(Equation 5)} \] Subtract Equation 4 from Equation 5: \[ (5a + b) - (3a + b) = 5 - 3 \quad \Rightarrow \quad 2a = 2 \quad \Rightarrow \quad a = 1 \] Plugging \( a = 1 \) into Equation 4: \[ 3(1) + b = 3 \quad \Rightarrow \quad b = 0 \] Plugging \( a = 1 \) and \( b = 0 \) into Equation 1: \[ 1 + 0 + c = -1 \quad \Rightarrow \quad c = -2 \] **Final \( n^{\text{th}} \) Term Formula:** \[ a_n = n^2 - 2 \] **Verification:** \[ \begin{align*} n = 1: & \quad 1^2 - 2 = -1 \\ n = 2: & \quad 2^2 - 2 = 2 \\ n = 3: & \quad 3^2 - 2 = 7 \\ n = 4: & \quad 4^2 - 2 = 14 \\ n = 5: & \quad 5^2 - 2 = 23 \\ \end{align*} \] All terms match the given sequence. ### b) Determining the Position of the Term with Value 62 We need to find the value of \( n \) such that: \[ a_n = n^2 - 2 = 62 \] **Solving for \( n \):** \[ n^2 - 2 = 62 \\ n^2 = 64 \\ n = \sqrt{64} \\ n = 8 \quad \text{(since position \( n \) is positive)} \] **Conclusion:** The term with the value **62** appears in the **8th** position of the sequence. --- ### **Final Answers:** **a)** The \( n^{\text{th}} \) term is \( a_n = n^2 - 2 \). **b)** The term with value 62 appears in position 8.