Pregunta
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) \( \quad 2 ; 8 ; 14 ; 20 ; \ldots \ldots \). \( \begin{array}{ll}\text { (j) } & 2 ; 0 ;-2 ;-4 ; \ldots \ldots .\end{array} \) (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,\( 25 ; 0,39 ; 0,53 ; \ldots \)
Ask by Simmons Kelley. in South Africa
Mar 06,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
**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 \( a_{15} = 10 + (15-1) \cdot (-3) = 10 - 42 = -32 \).
Solución
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: \( a_n = 10 + (n-1) \cdot (-3) \)
Respondido por UpStudy AI y revisado por un tutor profesional
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Simplifique esta solución Extra Insights
Let’s dive into two of the number patterns you provided! (a) The pattern 5, 7, 9, 11 follows a simple rule: you add 2 to each term to get the next one. The constant difference here is 2. The next three terms would be 13, 15, and 17. For the 15th term, you can use the general rule \( n \)th term = \( 5 + (n - 1) \times 2 \). Plugging in 15 gives you: \( 5 + (15 - 1) \times 2 = 33 \). (c) In the sequence 15, 25, 35, the rule involves adding 10 each time. The constant difference is 10. Next up, we can expect 45, 55, and 65. The 120th term can be found using the same formula: \( n \)th term = \( 15 + (n - 1) \times 10 \). So, for \( n = 120 \): \( 15 + (120 - 1) \times 10 = 1,185 \).
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