2.3 The arithmetic sequence, \( 2 ;-1 ;-4 ; \) forms the first differences of a certain quadratic sequence. If the first term of the quadratic pattern is equal to -1 . Determine the general term of the quadratic sequence.

To solve this problem, we start by analyzing the arithmetic sequence \( 2, -1, -4 \), which represents the first differences of a quadratic sequence. The first differences indicate how the values of the quadratic sequence change from one term to the next. 1. The first differences are \( 2 \) (from \( -1 \) to \( -4 \)); \( -1 \) (from \( 2 \) to \( -1 \)); and \( -4 \) (from \( -1 \) to the next term). The first differences themselves create a new sequence formed from a constant second difference because quadratic sequences have constant second differences. 2. Let's denote the quadratic sequence as \( a_n = An^2 + Bn + C \). We know the first term \( a_1 = -1 \) (when \( n=1 \)). 3. From our first differences, we know that: - \( a_2 - a_1 = 2 \) - \( a_3 - a_2 = -1 \) 4. This leads us to the equations: - For the first term: \( a_1 = A(1)^2 + B(1) + C = -1 \) - For the second term: \( a_2 = a_1 + 2 = -1 + 2 = 1 \) - For the third term: \( a_3 = a_2 - 1 = 1 - 1 = 0 \) 5. Now we have: - \( a_1 = A + B + C = -1 \) - \( a_2 = 4A + 2B + C = 1 \) - \( a_3 = 9A + 3B + C = 0 \) 6. By solving these equations, we can find values of A, B, and C. After solving, we find that: - \( A = -1 \) - \( B = 2 \) - \( C = -2 \) Thus, the general term of the quadratic sequence can be expressed as: \[ a_n = -n^2 + 2n - 2 \] So, the final result for the general term of the quadratic sequence is: \[ a_n = -n^2 + 2n - 2 \]