(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 find the general term of the sequence with a constant second difference of 8, we can express it as a quadratic equation. The general form of such a sequence can be given as \( a_n = An^2 + Bn + C \), where \( A \), \( B \), and \( C \) are constants to be determined. Since the second difference is constant and equal to 8, we know that \( 2A = 8 \), leading us to \( A = 4 \). Now we substitute the known terms into our equation: 1. For \( n = 6 \), \( a_6 = 172 \): \[ 4(6^2) + B(6) + C = 172 \] \[ 144 + 6B + C = 172 \] \[ 6B + C = 28 \] (Equation 1) 2. For \( n = 10 \), \( a_{10} = 452 \): \[ 4(10^2) + B(10) + C = 452 \] \[ 400 + 10B + C = 452 \] \[ 10B + C = 52 \] (Equation 2) Now we have a system of equations: - From Equation 1: \( 6B + C = 28 \) - From Equation 2: \( 10B + C = 52 \) Subtract these equations to eliminate \( C \): \[ (10B + C) - (6B + C) = 52 - 28 \] \[ 4B = 24 \implies B = 6 \] Now, substituting \( B = 6 \) back into Equation 1: \[ 6(6) + C = 28 \implies 36 + C = 28 \implies C = -8 \] Thus, we have found \( A \), \( B \), and \( C \): - \( A = 4 \) - \( B = 6 \) - \( C = -8 \) The general term of the sequence is given by: \[ a_n = 4n^2 + 6n - 8 \] And there you have the general term!