Exercise 1: (answers on page 340) 1) In the following sequences determine the: a) \( 12^{\text {ht }} \) term of \( 5 ; 12 ; 19 ; \ldots \) e) \( 18^{\text {th }} \) term of \( 4 ; 4+2 x ; 4+4 x ; \ldots \) b) \( 11^{\text {th term }} \) of \( -3 ; 6 ;-12 ; \ldots \) f) \( 25^{\text {th }} \) term of \( a ; a b ; a b^{2} ; \ldots \) c) \( 8^{\text {th }} \) term of \( 24 ; 6 ; \frac{3}{2} ; \ldots \) 2) Which term of the sequence: a) \( 5: 11: 17 ; \ldots \) is equal to 101 ? c) \( 5 x ; 20 x ; 80 x ; \ldots \) is equal to \( 20480 x \) ? b) \( 10: \frac{21}{2}: 11 \div \). is equal to \( \frac{41}{2} \) ? d) \( 21 ;-7 ; \frac{7}{3} ; \ldots \) is equal to \( \frac{7}{243} \) ? 3) Determine an expression for the \( \mathrm{n}^{\text {th }} \) term of the following sequence if the: a) \( 4^{\oplus} \) term is 24 and the \( 7^{\text {th }} \) term is 192 in a geometric sequence. c) \( 6^{\text {th }} \) term is 13 and the \( 14^{\text {th }} \) term is 33 in an arithmetic sequence. b) \( 2^{\text {ax }} \) term is 1 and the \( 9^{\text {th }} \) term is -34 in an anithmetic sequence. d) \( 5^{\text {th }} \) term is 9 and the \( 9^{\text {th }} \) term is 729 and the sequence contains alternating positive and

Let's solve the problems step by step. ### Exercise 1 #### 1) Determine the specified terms of the sequences: **a)** The sequence is \( 5, 12, 19, \ldots \) This is an arithmetic sequence where the first term \( a = 5 \) and the common difference \( d = 12 - 5 = 7 \). The \( n^{\text{th}} \) term of an arithmetic sequence is given by: \[ T_n = a + (n-1)d \] For the \( 12^{\text{th}} \) term: \[ T_{12} = 5 + (12-1) \cdot 7 = 5 + 11 \cdot 7 = 5 + 77 = 82 \] **b)** The sequence is \( -3, 6, -12, \ldots \) This is a geometric sequence where the first term \( a = -3 \) and the common ratio \( r = \frac{6}{-3} = -2 \). The \( n^{\text{th}} \) term of a geometric sequence is given by: \[ T_n = a \cdot r^{n-1} \] For the \( 11^{\text{th}} \) term: \[ T_{11} = -3 \cdot (-2)^{11-1} = -3 \cdot (-2)^{10} = -3 \cdot 1024 = -3072 \] **c)** The sequence is \( 24, 6, \frac{3}{2}, \ldots \) This is a geometric sequence where the first term \( a = 24 \) and the common ratio \( r = \frac{6}{24} = \frac{1}{4} \). For the \( 8^{\text{th}} \) term: \[ T_{8} = 24 \cdot \left(\frac{1}{4}\right)^{8-1} = 24 \cdot \left(\frac{1}{4}\right)^{7} = 24 \cdot \frac{1}{16384} = \frac{24}{16384} = \frac{3}{2048} \] **e)** The sequence is \( 4, 4 + 2x, 4 + 4x, \ldots \) This is an arithmetic sequence where the first term \( a = 4 \) and the common difference \( d = 2x \). For the \( 18^{\text{th}} \) term: \[ T_{18} = 4 + (18-1) \cdot 2x = 4 + 34x \] **f)** The sequence is \( a, ab, ab^2, \ldots \) This is a geometric sequence where the first term \( a = a \) and the common ratio \( r = b \). For the \( 25^{\text{th}} \) term: \[ T_{25} = a \cdot b^{25-1} = a \cdot b^{24} \] ### 2) Which term of the sequence is equal to the specified value? **a)** The sequence is \( 5, 11, 17, \ldots \) This is an arithmetic sequence where \( a = 5 \) and \( d = 6 \). To find \( n \) such that \( T_n = 101 \): \[ T_n = 5 + (n-1) \cdot 6 = 101 \] \[ (n-1) \cdot 6 = 96 \implies n-1 = 16 \implies n = 17 \] **b)** The sequence is \( 10, \frac{21}{2}, 11, \ldots \) This is an arithmetic sequence where \( a = 10 \) and \( d = \frac{21}{2} - 10 = \frac{1}{2} \). To find \( n \) such that \( T_n = \frac{41}{2} \): \[ T_n = 10 + (n-1) \cdot \frac{1}{2} = \frac{41}{2} \] \[ (n-1) \cdot \frac{1}{2} = \frac{21}{2} \implies n-1 = 21 \implies n = 22 \] **c)** The sequence is \( 5x, 20x, 80x, \ldots \) This is a geometric sequence where \( a = 5x \) and \( r = 4 \). To find \( n \) such that \( T_n = 20480x \): \[ T_n = 5x \cdot 4^{n-1} = 20480x \] \[ 4^{n-1} = \frac{20480}{5} = 4096 \implies 4^{n-1} = 4^6 \implies n-1 = 6 \implies n = 7 \] **d)** The sequence is \( 21, -7, \frac{7}{3}, \ldots \) This is a geometric sequence where \( a = 21 \) and \( r = -\frac{1}{3} \). To find \( n \) such that \( T_n = \frac{7}{243} \): \[ T_n = 21 \cdot \left(-\frac{1}{3}\right)^{n-1} = \frac{7}{243} \] \[ \left(-\frac{1}{3}\right)^{n-1} = \frac{1}{729} \implies n-1 = 6 \implies n = 7 \] ### 3) Determine an expression for the \( n^{\text{th}} \) term of the sequences: **a)** Given \( T_4 = 24 \) and \( T_7 = 192 \) in a geometric sequence. Let \( a \) be the first term and \( r \) the common ratio: \[ T_4 = a \cdot r^3 = 24 \] \[ T_7 = a \cdot r^6 = 192 \] Dividing the second equation by the first: \[ \frac{T_7}{T_4} = \frac{192}{24} = 8 \implies r^3 = 8 \implies r = 2 \] Substituting \( r \) back: \[ a \cdot 2^3 = 24 \implies a \cdot 8 = 24 \implies a = 3 \] Thus, the \( n^{\text{th}} \) term is: \[ T_n =