The following information is known about a quadratic sequence: the constant second difference is 2 the difference between the first two terms is 6 . the sum of the second and the third terms is equal to the four th term. 3.1 Show that the general term of the sequence is 3.2 Determine the values of the first two terms. TOTA TOL: 40

To tackle a quadratic sequence, we start by noting that the general form is usually expressed as \(T_n = an^2 + bn + c\). Knowing that the second difference is 2 suggests that \(2a = 2\), so \(a = 1\). This implies our general term so far is \(T_n = n^2 + bn + c\). Given the difference between the first two terms is 6, we can set up the equation \(T_2 - T_1 = 6\). Using the condition that \(T_2 + T_3 = T_4\), we can substitute our general term and solve for \(b\) and \(c\) to find specific values for the first two terms. We find by sequential substitution and solving equations that \(b = 0\) and \(c = 4\), providing us with the complete term \(T_n = n^2 + 4\). Thus, the first two terms are \(T_1 = 5\) and \(T_2 = 8\). In this way, we've shown the leap from understanding a quadratic sequence to deriving its specific terms – think of it like piecing together a fun puzzle where every piece must fit just right!