\begin{tabular}{l}\hline ACTIVITIESIASSESSMENTS \ \hline - Classwork/Homework \ 1. Consider the sequence: $$ 12 ; 4 ; 14 ; 7 ; 18 ; 10 ; \ldots $$ \ 1.1 Write down the next TWO terms if the given pattern continues. \ 1.2 Calculate the value of the $$ 50^{ ext {th }} $$ term of the sequence. \ 1.3 Write down the value of $$ 131^{ ext {th }} $$ term of the sequence. \ 2. The following sequence is a combination of an arithmetic and a geometric sequence: \ $$ 3 ; 3 ; 9 ; 6 ; 15 ; 12 ; \ldots $$ \ 2.1 Write down the next TWO terms. \ 2.2 Calculate $$ T_{22}-T_{21} $$. \ 2.3 Prove that ALL the terms of this infinite sequence will be divisible by 3 . \end{tabular}

Question

\begin{tabular}{l}\hline ACTIVITIESIASSESSMENTS \ \hline - Classwork/Homework \ 1. Consider the sequence: $$ 12 ; 4 ; 14 ; 7 ; 18 ; 10 ; \ldots $$ \ 1.1 Write down the next TWO terms if the given pattern continues. \ 1.2 Calculate the value of the $$ 50^{	ext {th }} $$ term of the sequence. \ 1.3 Write down the value of $$ 131^{	ext {th }} $$ term of the sequence. \ 2. The following sequence is a combination of an arithmetic and a geometric sequence: \ $$ 3 ; 3 ; 9 ; 6 ; 15 ; 12 ; \ldots $$ \ 2.1 Write down the next TWO terms. \ 2.2 Calculate $$ T_{22}-T_{21} $$. \ 2.3 Prove that ALL the terms of this infinite sequence will be divisible by 3 . \end{tabular}

UpStudy AI · Accepted Answer

Sure, let's work through each of the problems step by step.

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## **1. Sequence Analysis**

Given Sequence: **12 ; 4 ; 14 ; 7 ; 18 ; 10 ; …**

### **1.1 Next TWO Terms**

**Pattern Identification:**

- **Odd-Positioned Terms (1st, 3rd, 5th, ...):**  
  12, 14, 18, ...  
  - **Differences:** +2 (14 - 12), +4 (18 - 14), ...  
  - **Pattern:** The difference increases by 2 each time.  
  - **Next Difference:** +6  
  - **Next Odd Term:** 18 + 6 = **24**

- **Even-Positioned Terms (2nd, 4th, 6th, ...):**  
  4, 7, 10, ...  
  - **Differences:** +3 each time  
  - **Next Even Term:** 10 + 3 = **13**

**Next TWO Terms:** **24** and **13**

---

### **1.2 Value of the $$ 50^{\text{th}} $$ Term**

**Determining the Position:**

- **50 is Even**, so it follows the even-positioned pattern.

**Even-Term Formula:**
$$ a_n = 3k + 1 $$
where $$ k = \frac{n}{2} $$

$$ k = \frac{50}{2} = 25 $$
$$ a_{50} = 3 \times 25 + 1 = 75 + 1 = 76 $$

**$$ 50^{\text{th}} $$ Term:** **76**

---

### **1.3 Value of the $$ 131^{\text{th}} $$ Term**

**Determining the Position:**

- **131 is Odd**, so it follows the odd-positioned pattern.

**Odd-Term Formula:**
$$ a_n = \frac{n^2 + 47}{4} $$

$$ a_{131} = \frac{131^2 + 47}{4} $$
$$ = \frac{17161 + 47}{4} $$
$$ = \frac{17208}{4} $$
$$ = 4302 $$

**$$ 131^{\text{th}} $$ Term:** **4302**

---

## **2. Combined Arithmetic and Geometric Sequence**

Given Sequence: **3 ; 3 ; 9 ; 6 ; 15 ; 12 ; …**

### **2.1 Next TWO Terms**

**Pattern Identification:**

- **Odd-Positioned Terms (1st, 3rd, 5th, ...):**  
  3, 9, 15, ...  
  - **Differences:** +6 each time  
  - **Next Odd Term:** 15 + 6 = **21**

- **Even-Positioned Terms (2nd, 4th, 6th, ...):**  
  3, 6, 12, ...  
  - **Pattern:** Each term is multiplied by 2  
  - **Next Even Term:** 12 × 2 = **24**

**Next TWO Terms:** **21** and **24**

---

### **2.2 Calculate $$ T_{22} - T_{21} $$**

**Determining the Terms:**

- **$$ T_{21} $$ (Odd Position):**
  $$ a_k = 3 + 6(k - 1) $$
  $$ k = \frac{21 + 1}{2} = 11 $$
  $$ a_{21} = 3 + 6(11 - 1) = 3 + 60 = 63 $$

- **$$ T_{22} $$ (Even Position):**
  $$ a_k = 3 \times 2^{k-1} $$
  $$ k = \frac{22}{2} = 11 $$
  $$ a_{22} = 3 \times 2^{10} = 3 \times 1024 = 3072 $$

**Calculation:**
$$ T_{22} - T_{21} = 3072 - 63 = 3009 $$

**Result:** **3009**

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### **2.3 Proof that ALL Terms are Divisible by 3**

**Odd-Positioned Terms:**

- **Formula:**  
  $$ a_k = 3 + 6(k - 1) $$
  $$ = 3(1 + 2(k - 1)) $$
  - Clearly, $$ a_k $$ is a multiple of 3.

**Even-Positioned Terms:**

- **Formula:**  
  $$ a_k = 3 \times 2^{k-1} $$
  - Regardless of the value of $$ k $$, $$ a_k $$ is $$ 3 $$ multiplied by an integer $$ 2^{k-1} $$, making it a multiple of 3.

**Conclusion:**  
All terms, whether odd or even-positioned, are multiples of **3**.

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## **Summary of Answers**

1. **Sequence 12; 4; 14; 7; 18; 10; …**
   - **1.1 Next TWO Terms:** 24 and 13
   - **1.2 $$ 50^{\text{th}} $$ Term:** 76
   - **1.3 $$ 131^{\text{th}} $$ Term:** 4302

2. **Sequence 3; 3; 9; 6; 15; 12; …**
   - **2.1 Next TWO Terms:** 21 and 24
   - **2.2 $$ T_{22} - T_{21} $$:** 3009
   - **2.3 Proof of Divisibility by 3:** All terms are divisible by 3.

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If you have any further questions or need additional explanations, feel free to ask!
**Sequence Analysis:** 1. **Next TWO Terms:** 24 and 13 2. **50th Term:** 76 3. **131st Term:** 4302 **Combined Sequence:** 1. **Next TWO Terms:** 21 and 24 2. **$$ T_{22} - T_{21} $$:** 3009 3. **Proof:** All terms are divisible by 3.

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