The first four terms of a quadratic number pattern are \( -1 ; 2 ; 9 ; 20 \) 2.1.1 Determine the general term of the quadratic number pattern.

To find the general term of the quadratic pattern, we start by observing the terms: \( -1, 2, 9, 20 \). We calculate the first differences: \[ 2 - (-1) = 3, \] \[ 9 - 2 = 7, \] \[ 20 - 9 = 11. \] Now, the second differences are: \[ 7 - 3 = 4, \] \[ 11 - 7 = 4. \] Since the second differences are constant, the sequence is quadratic, taking the form \( an^2 + bn + c \). Using the first four terms, we set up the following equations: 1. For \( n=1 \): \( a(1)^2 + b(1) + c = -1 \) ⇒ \( a + b + c = -1 \) 2. For \( n=2 \): \( a(2)^2 + b(2) + c = 2 \) ⇒ \( 4a + 2b + c = 2 \) 3. For \( n=3 \): \( a(3)^2 + b(3) + c = 9 \) ⇒ \( 9a + 3b + c = 9 \) Now we have a system of three equations: 1. \( a + b + c = -1 \) (1) 2. \( 4a + 2b + c = 2 \) (2) 3. \( 9a + 3b + c = 9 \) (3) Subtract equations (1) from (2) and (2) from (3) to eliminate \( c \): From (2) - (1): \[ 3a + b = 3 \] (4) From (3) - (2): \[ 5a + b = 7 \] (5) Now, subtract equation (4) from (5): \[ (5a + b) - (3a + b) = 7 - 3 \] \[ 2a = 4 \] \[ a = 2 \] Now substitute \( a = 2 \) back into equation (4): \[ 3(2) + b = 3 \] \[ 6 + b = 3 \] \[ b = -3 \] Finally, substitute \( a \) and \( b \) into equation (1): \[ 2 - 3 + c = -1 \] \[ -1 + c = -1 \] \[ c = 0 \] Thus, the general term of the quadratic number pattern is: \[ T(n) = 2n^2 - 3n \]