QUESTION 3 3.1 The general term of: \( 5 ; 12 ; 29 ; 48 ; 77 ; \ldots \) is \( T_{n}=3 n^{2}+2 \) Is this statement true or false? Motivate your answer. 3.2 The first term of a linear pattern is 92 and the constant different is -4 3.2 1 Write down the value of the \( T_{2} \) and \( T_{3} \) of this number pattern. 3.2.2 Determine an expression for \( T_{n} \) 3.2.3 Determine the value of \( T_{18} \) 3.3 The \( n^{\text {th }} \) term of a number pattern is as follows. - \( T_{n}=-5 n-4 \) if the \( n \) is an even number. - \( T_{n}=-n^{2}+6 \) if \( n \) is an odd number. 3.3.1 Determine the value of: \( T_{6}+T_{7} \) 3.3.2 Determine the value of k if \( T_{k}=-219 \)

Let's go through each part of the question step by step. ### 3.1 **Statement:** The general term of: \( 5 ; 12 ; 29 ; 48 ; 77 ; \ldots \) is \( T_{n}=3 n^{2}+2 \). To verify this statement, we can calculate the first few terms using the proposed formula \( T_n = 3n^2 + 2 \): - For \( n = 1 \): \[ T_1 = 3(1^2) + 2 = 3(1) + 2 = 3 + 2 = 5 \] - For \( n = 2 \): \[ T_2 = 3(2^2) + 2 = 3(4) + 2 = 12 + 2 = 14 \quad \text{(not 12)} \] - For \( n = 3 \): \[ T_3 = 3(3^2) + 2 = 3(9) + 2 = 27 + 2 = 29 \] - For \( n = 4 \): \[ T_4 = 3(4^2) + 2 = 3(16) + 2 = 48 + 2 = 50 \quad \text{(not 48)} \] - For \( n = 5 \): \[ T_5 = 3(5^2) + 2 = 3(25) + 2 = 75 + 2 = 77 \] Since \( T_2 \) does not equal 12 and \( T_4 \) does not equal 48, the statement is **false**. ### 3.2 **Given:** The first term of a linear pattern is 92 and the constant difference is -4. #### 3.2.1 To find \( T_2 \) and \( T_3 \): - \( T_1 = 92 \) - \( T_2 = T_1 + (-4) = 92 - 4 = 88 \) - \( T_3 = T_2 + (-4) = 88 - 4 = 84 \) Thus, \( T_2 = 88 \) and \( T_3 = 84 \). #### 3.2.2 The general term \( T_n \) of a linear pattern can be expressed as: \[ T_n = T_1 + (n-1) \cdot d \] where \( d \) is the common difference. Here, \( T_1 = 92 \) and \( d = -4 \): \[ T_n = 92 + (n-1)(-4) = 92 - 4(n-1) = 92 - 4n + 4 = 96 - 4n \] Thus, the expression for \( T_n \) is: \[ T_n = 96 - 4n \] #### 3.2.3 To find \( T_{18} \): \[ T_{18} = 96 - 4(18) = 96 - 72 = 24 \] ### 3.3 **Given:** - \( T_{n}=-5 n-4 \) if \( n \) is an even number. - \( T_{n}=-n^{2}+6 \) if \( n \) is an odd number. #### 3.3.1 To find \( T_6 + T_7 \): - For \( T_6 \) (even): \[ T_6 = -5(6) - 4 = -30 - 4 = -34 \] - For \( T_7 \) (odd): \[ T_7 = -7^2 + 6 = -49 + 6 = -43 \] Thus, \[ T_6 + T_7 = -34 + (-43) = -77 \] #### 3.3.2 To find \( k \) such that \( T_k = -219 \): - If \( k \) is even: \[ -5k - 4 = -219 \implies -5k = -219 + 4 \implies -5k = -215 \implies k = \frac{215}{5} = 43 \] - If \( k \) is odd: \[ -k^2 + 6 = -219 \implies -k^2 = -219 - 6 \implies -k^2 = -225 \implies k^2 = 225 \implies k = 15 \text{ or } k = -15 \] Since \( k \) must be a positive integer, we have \( k = 15 \) (odd). Thus, the possible values of \( k \) are \( 43 \) (even) and \( 15 \) (odd).