3.1 A quadratic number pattern \( T_{n}=a n^{2}+b n+c \) has a third term equal to -1 , while the first differences of the quadratic sequence are given by: \( -12 ;-8 ;-4 \) 3.1.1 Write down the values of the first four terms of the quadratic sequence.

To find the first four terms of the quadratic sequence, we start with the first differences provided: \( -12, -8, -4 \). 1. The second difference of any quadratic sequence is constant. Here, the second differences can be calculated as follows: - \( -8 - (-12) = 4 \) - \( -4 - (-8) = 4 \) This constant second difference tells us that the coefficient \( a \) in the quadratic expression \( T_n = a n^2 + bn + c \) equals \( 2 \) (since \( 2a = 4 \)). 2. Now, to find the first differences and the original terms: - Let the first term be \( T_1 = T_1 \), - The second term is \( T_2 = T_1 - 12 \), - The third term is \( T_3 = T_1 - 12 - 8 = T_1 - 20 \), - The fourth term is \( T_4 = T_1 - 12 - 8 - 4 = T_1 - 24 \). 3. We know the third term is \( T_3 = -1 \): - Setting up the equation: \( T_1 - 20 = -1 \). - Solving for \( T_1 \) gives us \( T_1 = 19 \). Now we can find: - \( T_1 = 19 \) - \( T_2 = 19 - 12 = 7 \) - \( T_3 = -1 \) - \( T_4 = 19 - 24 = -5 \) Therefore, the first four terms of the quadratic sequence are \( 19, 7, -1, -5 \).