(a) Find the general term \( \left(T_{n}\right) \) of each of the following linear number patterns: (1) \( 5 ; 8 ; 11 ; 14 ; \ldots \) (2) \( 1 ;-1 ;-3 ;-5 ; \ldots \) (b) The sequence \( x-1 ; 2 x-3 ; x+6 \) is a linear pattern. Determine (1) the value of \( x \). (2) the general ferm \( \left(T_{n}\right) \). (c) Determine the general 1erm of each of the following quadratic number patterns: (1) \( 2 ; 3 ; 8 ; 17 ; 30 ; \ldots \) (2) \( -5 ;-5 ;-3 ; 1 ; \ldots \) (3) \( 20 ; 14 ; 10 ; 8 ; \ldots \) (4) \( 6 ; 7: 5 ; 0 ; \ldots \)

Let's solve the problem step by step. ### (a) Finding the general term \( T_n \) of linear number patterns #### (1) Sequence: \( 5, 8, 11, 14, \ldots \) To find the general term of this sequence, we first identify the first term and the common difference. - First term \( a = 5 \) - Common difference \( d = 8 - 5 = 3 \) The general term of a linear sequence can be expressed as: \[ T_n = a + (n - 1)d \] Substituting the values: \[ T_n = 5 + (n - 1) \cdot 3 = 5 + 3n - 3 = 3n + 2 \] #### (2) Sequence: \( 1, -1, -3, -5, \ldots \) Again, we identify the first term and the common difference. - First term \( a = 1 \) - Common difference \( d = -1 - 1 = -2 \) Using the same formula for the general term: \[ T_n = a + (n - 1)d \] Substituting the values: \[ T_n = 1 + (n - 1)(-2) = 1 - 2n + 2 = -2n + 3 \] ### (b) The sequence \( x - 1, 2x - 3, x + 6 \) #### (1) Determine the value of \( x \) To find \( x \), we need to ensure that the sequence is linear. This means the difference between consecutive terms should be constant. Calculating the differences: - First difference: \( (2x - 3) - (x - 1) = 2x - 3 - x + 1 = x - 2 \) - Second difference: \( (x + 6) - (2x - 3) = x + 6 - 2x + 3 = -x + 9 \) Setting the two differences equal: \[ x - 2 = -x + 9 \] Solving for \( x \): \[ 2x = 11 \implies x = \frac{11}{2} \] #### (2) Determine the general term \( T_n \) Now substituting \( x = \frac{11}{2} \) into the terms: - First term: \( \frac{11}{2} - 1 = \frac{9}{2} \) - Second term: \( 2 \cdot \frac{11}{2} - 3 = 11 - 3 = 8 \) - Third term: \( \frac{11}{2} + 6 = \frac{11}{2} + \frac{12}{2} = \frac{23}{2} \) Now we can find the common difference: - First difference: \( 8 - \frac{9}{2} = \frac{16}{2} - \frac{9}{2} = \frac{7}{2} \) - Second difference: \( \frac{23}{2} - 8 = \frac{23}{2} - \frac{16}{2} = \frac{7}{2} \) The common difference is \( \frac{7}{2} \). Using the general term formula: \[ T_n = a + (n - 1)d \] Where \( a = \frac{9}{2} \) and \( d = \frac{7}{2} \): \[ T_n = \frac{9}{2} + (n - 1) \cdot \frac{7}{2} = \frac{9}{2} + \frac{7n - 7}{2} = \frac{7n + 2}{2} \] ### (c) Finding the general term of quadratic number patterns #### (1) Sequence: \( 2, 3, 8, 17, 30, \ldots \) To find the general term, we first calculate the first differences: - \( 3 - 2 = 1 \) - \( 8 - 3 = 5 \) - \( 17 - 8 = 9 \) - \( 30 - 17 = 13 \) Now, calculate the second differences: - \( 5 - 1 = 4 \) - \( 9 - 5 = 4 \) - \( 13 - 9 = 4 \) Since the second differences are constant, this is a quadratic sequence. The general form is: \[ T_n = an^2 + bn + c \] Using the first three terms to set up equations: 1. \( a(1^2) + b(1) + c = 2 \) → \( a + b + c = 2 \) 2. \( a(2^2) + b(2) + c = 3 \) → \( 4a + 2b + c = 3 \) 3. \( a(3^2) + b(3) + c = 8 \) → \( 9a + 3b + c = 8 \) Now we can solve this system of equations. #### (2) Sequence: \( -5, -5, -3, 1, \ldots \) Calculating the first differences: - \( -5 - (-5) = 0 \) - \( -3 - (-5) = 2 \) - \( 1 - (-3) = 4 \) Calculating the second differences: - \( 2 - 0 = 2 \) - \( 4 - 2 = 2 \) This is also a quadratic sequence. Using the same form \( T_n = an^2 + bn + c \) and setting up equations: 1. \( a + b + c = -5 \) 2. \( 4a + 2b + c = -5 \) 3. \( 9a + 3b + c = -3 \) #### (3) Sequence: \( 20, 14, 10, 8, \ldots \) Calculating the first differences: - \( 14 - 20 = -6 \) - \( 10 - 14 = -4 \) - \( 8 - 10 = -2 \) Calculating the second differences: - \( -4 - (-6) = 2 \) - \( -2 - (-4) = 2 \) This is also a quadratic sequence. Using the same form \( T_n = an^2 + bn + c \) and setting up equations: 1. \( a + b + c = 20 \) 2. \( 4a + 2b + c = 14 \) 3. \( 9a + 3b + c = 10 \) #### (4) Sequence: