Chapter 1 Number Patterns, Sequences and Series (Exerci Sequences (progressions) Exercise 1: (answers on page 340) 1) In the following sequences determine the: (a) \( 12^{\text {hh }} \) term of \( 5 ; 12 ; 19 ; \ldots \) b) \( 11^{\text {th }} \) term of \( -3 ; 6 ;-12 ; \ldots \) d) \( 21^{31} \) term of \( 12 ; \frac{17}{2} ; 5 ; \ldots \) c) \( 8^{\text {ib }} \) term of \( 24 ; 6 ; \frac{3}{2} ; \ldots \) e) \( 18^{\text {ti }} \) term of \( 4 ; 4+2 x ; 4+4 x \) f) \( 25^{\text {th }} \) term of \( a ; a b ; a b^{2} ; \ldots \)

Sure, let's solve each part of Exercise 1 step by step. ### 1(a) \( 12^{\text{th}} \) term of the sequence \( 5 ; 12 ; 19 ; \ldots \) **Type of Sequence:** Arithmetic sequence **First Term (\( a_1 \)):** 5 **Common Difference (\( d \)):** \( 12 - 5 = 7 \) **Formula for the \( n^{th} \) term of an arithmetic sequence:** \[ a_n = a_1 + (n-1)d \] **Calculating the 12th term:** \[ a_{12} = 5 + (12-1) \times 7 = 5 + 77 = 82 \] **Answer:** \( a_{12} = 82 \) --- ### 1(b) \( 11^{\text{th}} \) term of the sequence \( -3 ; 6 ; -12 ; \ldots \) **Type of Sequence:** Geometric sequence **First Term (\( a_1 \)):** -3 **Common Ratio (\( r \)):** \( \frac{6}{-3} = -2 \) **Formula for the \( n^{th} \) term of a geometric sequence:** \[ a_n = a_1 \times r^{(n-1)} \] **Calculating the 11th term:** \[ a_{11} = -3 \times (-2)^{10} = -3 \times 1024 = -3072 \] **Answer:** \( a_{11} = -3072 \) --- ### 1(c) \( 8^{\text{th}} \) term of the sequence \( 24 ; 6 ; \frac{3}{2} ; \ldots \) **Type of Sequence:** Geometric sequence **First Term (\( a_1 \)):** 24 **Common Ratio (\( r \)):** \( \frac{6}{24} = \frac{1}{4} \) **Formula for the \( n^{th} \) term of a geometric sequence:** \[ a_n = a_1 \times r^{(n-1)} \] **Calculating the 8th term:** \[ a_8 = 24 \times \left(\frac{1}{4}\right)^7 = 24 \times \frac{1}{16384} = \frac{3}{2048} \] **Answer:** \( a_8 = \frac{3}{2048} \) --- ### 1(d) \( 21^{\text{st}} \) term of the sequence \( 12 ; \frac{17}{2} ; 5 ; \ldots \) **Type of Sequence:** Arithmetic sequence **First Term (\( a_1 \)):** 12 **Common Difference (\( d \)):** \( \frac{17}{2} - 12 = \frac{17}{2} - \frac{24}{2} = -\frac{7}{2} \) **Formula for the \( n^{th} \) term of an arithmetic sequence:** \[ a_n = a_1 + (n-1)d \] **Calculating the 21st term:** \[ a_{21} = 12 + (21-1) \times \left(-\frac{7}{2}\right) = 12 - 70 = -58 \] **Answer:** \( a_{21} = -58 \) --- ### 1(e) \( 18^{\text{th}} \) term of the sequence \( 4 ; 4 + 2x ; 4 + 4x ; \ldots \) **Type of Sequence:** Arithmetic sequence **First Term (\( a_1 \)):** 4 **Common Difference (\( d \)):** \( (4 + 2x) - 4 = 2x \) **Formula for the \( n^{th} \) term of an arithmetic sequence:** \[ a_n = a_1 + (n-1)d \] **Calculating the 18th term:** \[ a_{18} = 4 + (18-1) \times 2x = 4 + 34x \] **Answer:** \( a_{18} = 4 + 34x \) --- ### 1(f) \( 25^{\text{th}} \) term of the sequence \( a ; a b ; a b^{2} ; \ldots \) **Type of Sequence:** Geometric sequence **First Term (\( a_1 \)):** \( a \) **Common Ratio (\( r \)):** \( b \) **Formula for the \( n^{th} \) term of a geometric sequence:** \[ a_n = a_1 \times r^{(n-1)} \] **Calculating the 25th term:** \[ a_{25} = a \times b^{24} \] **Answer:** \( a_{25} = a \times b^{24} \) --- ### Summary of Answers 1. **(a)** \( a_{12} = 82 \) 2. **(b)** \( a_{11} = -3072 \) 3. **(c)** \( a_{8} = \frac{3}{2048} \) 4. **(d)** \( a_{21} = -58 \) 5. **(e)** \( a_{18} = 4 + 34x \) 6. **(f)** \( a_{25} = a \times b^{24} \)