(o) A certain sequence has a constant second diffierence of 8. The sixth term of the sequence is 172 and the tenth term is 452. Determine the general term.

To determine the general term of the sequence with a constant second difference of 8, follow these steps: 1. **Understand the Type of Sequence:** - A sequence with a constant second difference is a quadratic sequence. - The general form of a quadratic sequence is: \[ a_n = an^2 + bn + c \] 2. **Determine the Coefficient \(a\):** - The second difference is given by \(2a\). - Given that the second difference is 8: \[ 2a = 8 \implies a = 4 \] - So, the general form becomes: \[ a_n = 4n^2 + bn + c \] 3. **Use Given Terms to Find \(b\) and \(c\):** - **Sixth Term (\(a_6 = 172\)):** \[ 4(6)^2 + 6b + c = 172 \implies 144 + 6b + c = 172 \implies 6b + c = 28 \quad \text{(Equation 1)} \] - **Tenth Term (\(a_{10} = 452\)):** \[ 4(10)^2 + 10b + c = 452 \implies 400 + 10b + c = 452 \implies 10b + c = 52 \quad \text{(Equation 2)} \] - **Subtract Equation 1 from Equation 2:** \[ (10b + c) - (6b + c) = 52 - 28 \implies 4b = 24 \implies b = 6 \] - **Substitute \(b = 6\) into Equation 1:** \[ 6(6) + c = 28 \implies 36 + c = 28 \implies c = -8 \] 4. **Write the General Term:** \[ a_n = 4n^2 + 6n - 8 \] **Final Answer:** The general term is 4 n² plus six n minus eight. In symbols, aₙ = 4 n² + 6 n − 8