2.3 Is 100 a term in the sequence? Justify your answer.
The sum of the second and sixth terms of an arithmetic sequence is 4
The third term is 24 niore than the term.
3.1 Determine the first three terms in the sequence.
3.2 Determine a formula for the term of the sequence.
3.3 Determine the term in the sequence.
3.4 Is 72 a term in the sequence? Justify your answer.
Which term in the sequence has a value of 8 ?
Consider the sequence 13,

Ask by Medina Cummings. in South Africa

Jan 16,2025

Question

2.3 Is 100 a term in the sequence? Justify your answer.
The sum of the second and sixth terms of an arithmetic sequence is 4
The third term is 24 niore than the term.
3.1 Determine the first three terms in the sequence.
3.2 Determine a formula for the term of the sequence.
3.3 Determine the term in the sequence.
3.4 Is 72 a term in the sequence? Justify your answer.
Which term in the sequence has a value of 8 ?
Consider the sequence 13,

Ask by Medina Cummings. in South Africa

Jan 16,2025

UpStudy AI · Accepted Answer

Let's tackle each part of your arithmetic sequence problems step by step.

### **Given:**
- **Arithmetic Sequence Properties:**
  - The sum of the second and sixth terms is 4.
  - The third term is 24 less than the 11th term.

Let's denote:
- **First term (a₁):** $$ a $$
- **Common difference (d):** $$ d $$

---

### **1. Establishing the Sequence**

**a. Finding $$ a $$ and $$ d $$:**

- **Second term (a₂):** $$ a + d $$
- **Sixth term (a₆):** $$ a + 5d $$

Given:
$$ a₂ + a₆ = 4 $$
$$ (a + d) + (a + 5d) = 4 $$
$$ 2a + 6d = 4 $$
$$ a + 3d = 2 \quad \text{(Equation 1)} $$

- **Third term (a₃):** $$ a + 2d $$
- **Eleventh term (a₁₁):** $$ a + 10d $$

Given:
$$ a₃ = a₁₁ - 24 $$
$$ a + 2d = (a + 10d) - 24 $$
$$ a + 2d = a + 10d - 24 $$
$$ 2d = 10d - 24 $$
$$ -8d = -24 $$
$$ d = 3 $$

Substitute $$ d = 3 $$ into **Equation 1**:
$$ a + 3(3) = 2 $$
$$ a + 9 = 2 $$
$$ a = -7 $$

**Thus, the general term (n-th term) of the sequence is:**
$$ a_n = a + (n-1)d $$
$$ a_n = -7 + (n-1)(3) $$
$$ a_n = 3n - 10 $$

---

### **2. Answering the Specific Questions**

#### **2.3. Is 100 a term in the sequence? Justify your answer.**

To determine if 100 is a term in the sequence, set $$ a_n = 100 $$ and solve for $$ n $$:

$$ 3n - 10 = 100 $$
$$ 3n = 110 $$
$$ n = \frac{110}{3} \approx 36.666... $$

**Conclusion:** Since $$ n $$ is not an integer, **100 is not a term in the sequence**.

---

### **3. Further Questions Based on the Sequence**

#### **3.1. Determine the first three terms in the sequence.**

Using $$ a = -7 $$ and $$ d = 3 $$:

- **First term (a₁):** $$ -7 $$
- **Second term (a₂):** $$ -7 + 3 = -4 $$
- **Third term (a₃):** $$ -7 + 2 \times 3 = -1 $$

**First three terms:** $$ -7, -4, -1 $$

#### **3.2. Determine a formula for the $$ n^{\text{th}} $$ term of the sequence.**

As established earlier:
$$ a_n = 3n - 10 $$

#### **3.3. Determine the $$ 17^{\text{th}} $$ term in the sequence.**

$$ a_{17} = 3(17) - 10 = 51 - 10 = 41 $$

**The 17th term is 41.**

#### **3.4. Is 72 a term in the sequence? Justify your answer.**

Set $$ a_n = 72 $$:
$$ 3n - 10 = 72 $$
$$ 3n = 82 $$
$$ n = \frac{82}{3} \approx 27.333... $$

**Conclusion:** Since $$ n $$ is not an integer, **72 is not a term in the sequence**.

---

### **4. Additional Sequence Analysis**

**Sequence Provided:** $$ 2; \frac{5}{3}, \frac{4}{3}, 1; \ldots $$

#### **Question: Which term in the sequence has a value of 8?**

**a. Identify the first term and common difference:**

- **First term (a₁):** $$ 2 $$
- **Second term (a₂):** $$ \frac{5}{3} $$

$$ d = a₂ - a₁ = \frac{5}{3} - 2 = \frac{5}{3} - \frac{6}{3} = -\frac{1}{3} $$

**b. General formula:**
$$ a_n = 2 + (n-1)\left(-\frac{1}{3}\right) $$
$$ a_n = 2 - \frac{n-1}{3} $$
$$ a_n = \frac{6}{3} - \frac{n-1}{3} $$
$$ a_n = \frac{7 - n}{3} $$

**c. Determine $$ n $$ when $$ a_n = 8 $$:**
$$ \frac{7 - n}{3} = 8 $$
$$ 7 - n = 24 $$
$$ -n = 17 $$
$$ n = -17 $$

**Conclusion:** Since $$ n = -17 $$ is not a valid term number (term numbers are positive integers), **8 is not a term in the sequence**.

---

### **5. Considering the Sequence $$ 13, 11, 9, 7, 5 $$**

This is an arithmetic sequence where:

- **First term (a₁):** $$ 13 $$
- **Common difference (d):** $$ -2 $$ (since each term decreases by 2)

**General term formula:**
$$ a_n = 13 + (n-1)(-2) $$
$$ a_n = 13 - 2n + 2 $$
$$ a_n = 15 - 2n $$

*If you have specific questions about this sequence, feel free to ask!*

---

Feel free to ask if you need further clarification or assistance with any of these problems!
**Simplified Answers:** 1. **Is 100 a term in the sequence?** - **Answer:** No, 100 is not a term in the sequence. 2. **First three terms:** - **Answer:** The first three terms are -7, -4, and -1. 3. **Formula for the $$ n^{\text{th}} $$ term:** - **Answer:** $$ a_n = 3n - 10 $$ 4. **17th term:** - **Answer:** The 17th term is 41. 5. **Is 72 a term in the sequence?** - **Answer:** No, 72 is not a term in the sequence. 6. **Which term equals 8 in the sequence $$ 2; \frac{5}{3}, \frac{4}{3}, 1; \ldots $$?** - **Answer:** 8 is not a term in the sequence. 7. **Sequence $$ 13, 11, 9, 7, 5 $$:** - **Formula for the $$ n^{\text{th}} $$ term:** $$ a_n = 15 - 2n $$ - **Note:** 8 is not a term in this sequence.

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