EXERCISE 1 For each of the following number patterns: (1) State the rule in words. (2) Write down the constant difference. (3) Write down the next three terms. (5) Calculate the 15 th term. (4) Determine the general rule ( $$ n $$th term). (a) $$ 5 ; 7 ; 9 ; 11 $$; $$ \qquad $$ (b) $$ 6 ; 11 ; 16 ; 21 $$; Calculate the 120 th term. (d) $$ \quad 6 ; 9 ; 12 ; 15 $$; $$ \qquad $$ (g) $$ \quad 2 ; 8 ; 14 ; 20 ; \ldots \ldots $$. $$ \begin{array}{ll} ext { (j) } & 2 ; 0 ;-2 ;-4 ; \ldots \ldots .\end{array} $$ (c) $$ 15 ; 25 ; 35 $$. (c) $$ 10 ; 12 ; 14 ; 16 ; $$ (h) $$ 6 ; 10 ; 14 ; 18 $$; (f) $$ 2 ; 6 ; 10 ; 14 $$; $$ \qquad $$ (k) $$ 10 ; 7 ; 4 ; 1 ; $$ (i) $$ 6 ; 15 ; 24 ; 33 $$; $$ \qquad $$ (p) $$ \quad 9 ; 4 ;-1 ;-6 $$; (n) $$ -7 ;-11 ;-15 $$; (1) $$ 18 ; 15 ; 12 $$. $$ \qquad $$ (p) $$ -12 ;-6 ; 0 ; 6 $$; (q) $$ 2 ;-8 ;-18 $$; $$ \qquad $$ (o) $$ -6 ;-11 ;-16 ; \ldots \ldots $$. (s) $$ 1 \frac{1}{2} ; 3 \frac{1}{2} ; 5 \frac{1}{2} ; \ldots $$. (t) $$ \frac{1}{2} ; 1 \frac{1}{2} ; 2 \frac{1}{2} ; 3 \frac{1}{2} ; \ldots $$ (u) $$ \quad \frac{1}{2} ; 2 ; 3 \frac{1}{2} ; 5 ; \ldots $$. (v) $$ \frac{1}{4} ; \frac{1}{2} ; \frac{3}{4} ; 1 ; \ldots $$. (w) $$ \frac{3}{5} ; \frac{2}{5} ; \frac{1}{5} ; 0 ; \ldots $$. (x) 0,$$ 25 ; 0,39 ; 0,53 ; \ldots $$

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EXERCISE 1 For each of the following number patterns: (1) State the rule in words. (2) Write down the constant difference. (3) Write down the next three terms. (5) Calculate the 15 th term. (4) Determine the general rule ( $$ n $$th term). (a) $$ 5 ; 7 ; 9 ; 11 $$; $$ \qquad $$ (b) $$ 6 ; 11 ; 16 ; 21 $$; Calculate the 120 th term. (d) $$ \quad 6 ; 9 ; 12 ; 15 $$; $$ \qquad $$ (g) $$ \quad 2 ; 8 ; 14 ; 20 ; \ldots \ldots $$. $$ \begin{array}{ll}	ext { (j) } & 2 ; 0 ;-2 ;-4 ; \ldots \ldots .\end{array} $$ (c) $$ 15 ; 25 ; 35 $$. (c) $$ 10 ; 12 ; 14 ; 16 ; $$ (h) $$ 6 ; 10 ; 14 ; 18 $$; (f) $$ 2 ; 6 ; 10 ; 14 $$; $$ \qquad $$ (k) $$ 10 ; 7 ; 4 ; 1 ; $$ (i) $$ 6 ; 15 ; 24 ; 33 $$; $$ \qquad $$ (p) $$ \quad 9 ; 4 ;-1 ;-6 $$; (n) $$ -7 ;-11 ;-15 $$; (1) $$ 18 ; 15 ; 12 $$. $$ \qquad $$ (p) $$ -12 ;-6 ; 0 ; 6 $$; (q) $$ 2 ;-8 ;-18 $$; $$ \qquad $$ (o) $$ -6 ;-11 ;-16 ; \ldots \ldots $$. (s) $$ 1 \frac{1}{2} ; 3 \frac{1}{2} ; 5 \frac{1}{2} ; \ldots $$. (t) $$ \frac{1}{2} ; 1 \frac{1}{2} ; 2 \frac{1}{2} ; 3 \frac{1}{2} ; \ldots $$ (u) $$ \quad \frac{1}{2} ; 2 ; 3 \frac{1}{2} ; 5 ; \ldots $$. (v) $$ \frac{1}{4} ; \frac{1}{2} ; \frac{3}{4} ; 1 ; \ldots $$. (w) $$ \frac{3}{5} ; \frac{2}{5} ; \frac{1}{5} ; 0 ; \ldots $$. (x) 0,$$ 25 ; 0,39 ; 0,53 ; \ldots $$

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Let's analyze each of the number patterns step by step. For each pattern, we will state the rule in words, identify the constant difference, write down the next three terms, calculate the 15th term, and determine the general rule (nth term).

### (a) $$ 5 ; 7 ; 9 ; 11 $$
1. **Rule in words**: Start at 5 and add 2 each time.
2. **Constant difference**: 2
3. **Next three terms**: 13, 15, 17
4. **15th term**: 
   - General rule: $$ a_n = 5 + (n-1) \cdot 2 $$
   - Calculate $$ a_{15} = 5 + (15-1) \cdot 2 = 5 + 28 = 33 $$
5. **General rule**: $$ a_n = 5 + 2(n-1) = 2n + 3 $$

### (b) $$ 6 ; 11 ; 16 ; 21 $$
1. **Rule in words**: Start at 6 and add 5 each time.
2. **Constant difference**: 5
3. **Next three terms**: 26, 31, 36
4. **120th term**: 
   - General rule: $$ a_n = 6 + (n-1) \cdot 5 $$
   - Calculate $$ a_{120} = 6 + (120-1) \cdot 5 = 6 + 595 = 601 $$
5. **General rule**: $$ a_n = 5n + 1 $$

### (c) $$ 15 ; 25 ; 35 $$
1. **Rule in words**: Start at 15 and add 10 each time.
2. **Constant difference**: 10
3. **Next three terms**: 45, 55, 65
4. **15th term**: 
   - General rule: $$ a_n = 15 + (n-1) \cdot 10 $$
   - Calculate $$ a_{15} = 15 + (15-1) \cdot 10 = 15 + 140 = 155 $$
5. **General rule**: $$ a_n = 10n + 5 $$

### (d) $$ 6 ; 9 ; 12 ; 15 $$
1. **Rule in words**: Start at 6 and add 3 each time.
2. **Constant difference**: 3
3. **Next three terms**: 18, 21, 24
4. **15th term**: 
   - General rule: $$ a_n = 6 + (n-1) \cdot 3 $$
   - Calculate $$ a_{15} = 6 + (15-1) \cdot 3 = 6 + 42 = 48 $$
5. **General rule**: $$ a_n = 3n + 3 $$

### (g) $$ 2 ; 8 ; 14 ; 20 $$
1. **Rule in words**: Start at 2 and add 6 each time.
2. **Constant difference**: 6
3. **Next three terms**: 26, 32, 38
4. **15th term**: 
   - General rule: $$ a_n = 2 + (n-1) \cdot 6 $$
   - Calculate $$ a_{15} = 2 + (15-1) \cdot 6 = 2 + 84 = 86 $$
5. **General rule**: $$ a_n = 6n - 4 $$

### (j) $$ 2 ; 0 ; -2 ; -4 $$
1. **Rule in words**: Start at 2 and subtract 2 each time.
2. **Constant difference**: -2
3. **Next three terms**: -6, -8, -10
4. **15th term**: 
   - General rule: $$ a_n = 2 + (n-1) \cdot (-2) $$
   - Calculate $$ a_{15} = 2 + (15-1) \cdot (-2) = 2 - 28 = -26 $$
5. **General rule**: $$ a_n = -2n + 4 $$

### (c) $$ 10 ; 12 ; 14 ; 16 $$
1. **Rule in words**: Start at 10 and add 2 each time.
2. **Constant difference**: 2
3. **Next three terms**: 18, 20, 22
4. **15th term**: 
   - General rule: $$ a_n = 10 + (n-1) \cdot 2 $$
   - Calculate $$ a_{15} = 10 + (15-1) \cdot 2 = 10 + 28 = 38 $$
5. **General rule**: $$ a_n = 2n + 8 $$

### (h) $$ 6 ; 10 ; 14 ; 18 $$
1. **Rule in words**: Start at 6 and add 4 each time.
2. **Constant difference**: 4
3. **Next three terms**: 22, 26, 30
4. **15th term**: 
   - General rule: $$ a_n = 6 + (n-1) \cdot 4 $$
   - Calculate $$ a_{15} = 6 + (15-1) \cdot 4 = 6 + 56 = 62 $$
5. **General rule**: $$ a_n = 4n + 2 $$

### (f) $$ 2 ; 6 ; 10 ; 14 $$
1. **Rule in words**: Start at 2 and add 4 each time.
2. **Constant difference**: 4
3. **Next three terms**: 18, 22, 26
4. **15th term**: 
   - General rule: $$ a_n = 2 + (n-1) \cdot 4 $$
   - Calculate $$ a_{15} = 2 + (15-1) \cdot 4 = 2 + 56 = 58 $$
5. **General rule**: $$ a_n = 4n - 2 $$

### (k) $$ 10 ; 7 ; 4 ; 1 $$
1. **Rule in words**: Start at 10 and subtract 3 each time.
2. **Constant difference**: -3
3. **Next three terms**: -2, -5, -8
4. **15th term**: 
   - General rule: $$ a_n = 10 + (n-1) \cdot (-3) $$

**Simplified Answers for Number Patterns:** 1. **Pattern (a):** - **Rule:** Start at 5 and add 2 each time. - **Constant Difference:** 2 - **Next Three Terms:** 13, 15, 17 - **15th Term:** 33 - **General Rule:** $$ a_n = 2n + 3 $$ 2. **Pattern (b):** - **Rule:** Start at 6 and add 5 each time. - **Constant Difference:** 5 - **Next Three Terms:** 26, 31, 36 - **120th Term:** 601 - **General Rule:** $$ a_n = 5n + 1 $$ 3. **Pattern (c):** - **Rule:** Start at 15 and add 10 each time. - **Constant Difference:** 10 - **Next Three Terms:** 45, 55, 65 - **15th Term:** 155 - **General Rule:** $$ a_n = 10n + 5 $$ 4. **Pattern (d):** - **Rule:** Start at 6 and add 3 each time. - **Constant Difference:** 3 - **Next Three Terms:** 18, 21, 24 - **15th Term:** 48 - **General Rule:** $$ a_n = 3n + 3 $$ 5. **Pattern (g):** - **Rule:** Start at 2 and add 6 each time. - **Constant Difference:** 6 - **Next Three Terms:** 26, 32, 38 - **15th Term:** 86 - **General Rule:** $$ a_n = 6n - 4 $$ 6. **Pattern (j):** - **Rule:** Start at 2 and subtract 2 each time. - **Constant Difference:** -2 - **Next Three Terms:** -6, -8, -10 - **15th Term:** -26 - **General Rule:** $$ a_n = -2n + 4 $$ 7. **Pattern (c):** - **Rule:** Start at 10 and add 2 each time. - **Constant Difference:** 2 - **Next Three Terms:** 18, 20, 22 - **15th Term:** 38 - **General Rule:** $$ a_n = 2n + 8 $$ 8. **Pattern (h):** - **Rule:** Start at 6 and add 4 each time. - **Constant Difference:** 4 - **Next Three Terms:** 22, 26, 30 - **15th Term:** 62 - **General Rule:** $$ a_n = 4n + 2 $$ 9. **Pattern (f):** - **Rule:** Start at 2 and add 4 each time. - **Constant Difference:** 4 - **Next Three Terms:** 18, 22, 26 - **15th Term:** 58 - **General Rule:** $$ a_n = 4n - 2 $$ 10. **Pattern (k):** - **Rule:** Start at 10 and subtract 3 each time. - **Constant Difference:** -3 - **Next Three Terms:** -2, -5, -8 - **15th Term:** - General rule: $$ a_n = 10 + (n-1) \cdot (-3) $$ **Note:** The 15th term for pattern (k) requires calculating $$ a_{15} = 10 + (15-1) \cdot (-3) = 10 - 42 = -32 $$.

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