Activity 4 4.1 Consider the linear sequence: \( 5 ; 8 ; 11 ; b: 17: \ldots \) 4.1.1 Write down the value of \( b \). 4.1.2 Determine the \( n^{t h} \) term of the sequence. 4.1.3 Calculate the value of the \( 15^{\oplus} \) term of the sequence. 4.1.4 Which term of the sequence is equal to 83 ? Activity 5 5.1 The first term of a linear number pattern is 92 and the constant difference is -4 . 5.1.1 Write down the values of the second and third terms of the pattern. 5.12 Determine an expression for the \( n^{\text {th }} \) term of the number patterm. 5.13 Determine the value of the eighteenth term. S.1.4 If \( T_{p}+T_{q}=0 \), determine the value of \( (p+q) \). TUTORLAL 2 Activity 1 1.1 Given the finite linear number pattern: \( 12 ; 17 ; 22 ; 22 ; \ldots ; 172 \). 1.1.1 Deternine the formula \( n^{\text {th }} \) term of the pattern. 1.1.2 Calculate the value of \( T_{12} \). 1.1.3 Deternine the number of terms in the pattern. 12 Given the first four terms of a linear pattern: \( 3: x: y: 30 \)

Let's solve the problems step by step. ### Activity 4 #### 4.1 Consider the linear sequence: \( 5 ; 8 ; 11 ; b ; 17 ; \ldots \) **4.1.1 Write down the value of \( b \).** The sequence is linear, meaning it has a constant difference. - The difference between the first two terms: \( 8 - 5 = 3 \) - The difference between the second and third terms: \( 11 - 8 = 3 \) - Therefore, the common difference \( d = 3 \). To find \( b \): \[ b = 11 + 3 = 14 \] **4.1.2 Determine the \( n^{th} \) term of the sequence.** The \( n^{th} \) term of a linear sequence can be expressed as: \[ T_n = a + (n-1)d \] where \( a \) is the first term and \( d \) is the common difference. Here, \( a = 5 \) and \( d = 3 \): \[ T_n = 5 + (n-1) \cdot 3 = 5 + 3n - 3 = 3n + 2 \] **4.1.3 Calculate the value of the \( 15^{th} \) term of the sequence.** Using the formula for the \( n^{th} \) term: \[ T_{15} = 3(15) + 2 = 45 + 2 = 47 \] **4.1.4 Which term of the sequence is equal to 83?** Set the \( n^{th} \) term equal to 83: \[ 3n + 2 = 83 \] \[ 3n = 81 \implies n = 27 \] ### Activity 5 #### 5.1 The first term of a linear number pattern is 92 and the constant difference is -4. **5.1.1 Write down the values of the second and third terms of the pattern.** - Second term: \( 92 - 4 = 88 \) - Third term: \( 88 - 4 = 84 \) **5.1.2 Determine an expression for the \( n^{th} \) term of the number pattern.** Using the formula for the \( n^{th} \) term: \[ T_n = a + (n-1)d \] where \( a = 92 \) and \( d = -4 \): \[ T_n = 92 + (n-1)(-4) = 92 - 4n + 4 = 96 - 4n \] **5.1.3 Determine the value of the eighteenth term.** Using the formula: \[ T_{18} = 96 - 4(18) = 96 - 72 = 24 \] **5.1.4 If \( T_{p} + T_{q} = 0 \), determine the value of \( (p + q) \).** Using the expression for \( T_n \): \[ T_p + T_q = (96 - 4p) + (96 - 4q) = 0 \] \[ 192 - 4(p + q) = 0 \implies 4(p + q) = 192 \implies p + q = 48 \] ### TUTORIAL 2 #### Activity 1 **1.1 Given the finite linear number pattern: \( 12 ; 17 ; 22 ; \ldots ; 172 \).** **1.1.1 Determine the formula \( n^{th} \) term of the pattern.** The common difference \( d \) is: \[ 17 - 12 = 5 \] Using the formula for the \( n^{th} \) term: \[ T_n = a + (n-1)d \] where \( a = 12 \) and \( d = 5 \): \[ T_n = 12 + (n-1) \cdot 5 = 12 + 5n - 5 = 5n + 7 \] **1.1.2 Calculate the value of \( T_{12} \).** Using the formula: \[ T_{12} = 5(12) + 7 = 60 + 7 = 67 \] **1.1.3 Determine the number of terms in the pattern.** Set \( T_n = 172 \): \[ 5n + 7 = 172 \] \[ 5n = 165 \implies n = 33 \] ### 12 Given the first four terms of a linear pattern: \( 3 ; x ; y ; 30 \) To find \( x \) and \( y \), we need to determine the common difference. Let \( d \) be the common difference: - From \( 3 \) to \( x \): \( x = 3 + d \) - From \( x \) to \( y \): \( y = x + d = 3 + 2d \) - From \( y \) to \( 30 \): \( 30 = y + d = 3 + 3d \) Setting up the equation: \[ 3 + 3d = 30 \implies 3d = 27 \implies d = 9 \] Now substituting back to find \( x \) and \( y \): \[ x = 3 + 9 = 12 \] \[ y = 3 + 2(9) = 21 \] Thus, the values are: - \( x = 12 \) - \( y = 21 \)