EXERCISE 6
Determine the first three terms of each of the following arithmetic
sequences of which:
(a) the 3 rd term of the sequence is 23 and the 26 th term is 230 .
(b) the 5 th term of the sequence is 19 and the 15 th term is 59 .
The 15 th and 3 rd terms of an arithmetic sequence are 100 and 28
respectively. Determine the 100 th term.

Question

UpStudy AI · Accepted Answer

Sure, let's tackle each part of Exercise 6 step by step.

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### **Arithmetic Sequences Refresher**

An **arithmetic sequence** is a sequence of numbers where the difference between consecutive terms is constant. This difference is known as the **common difference (d)**. The **nth term** of an arithmetic sequence can be calculated using the formula:

$$
a_n = a_1 + (n - 1) \cdot d
$$

Where:
- $$ a_n $$ = nth term
- $$ a_1 $$ = first term
- $$ d $$ = common difference
- $$ n $$ = term number

---

### **Part (a)**
**Given:**
- 3rd term ($$ a_3 $$) = 23
- 26th term ($$ a_{26} $$) = 230

**Find:** The first three terms of the sequence.

**Solution:**

1. **Set Up Equations:**

$$
\begin{cases}
a_3 = a_1 + 2d = 23 \quad \text{(1)}\
a_{26} = a_1 + 25d = 230 \quad \text{(2)}
\end{cases}
$$

2. **Subtract Equation (1) from Equation (2):**

$$
(a_1 + 25d) - (a_1 + 2d) = 230 - 23 \
23d = 207 \
d = \frac{207}{23} \
d = 9
$$

3. **Find $$ a_1 $$ Using Equation (1):**

$$
a_1 + 2 \cdot 9 = 23 \
a_1 + 18 = 23 \
a_1 = 23 - 18 \
a_1 = 5
$$

4. **Determine the First Three Terms:**

$$
\begin{align*}
a_1 &= 5 \
a_2 &= a_1 + d = 5 + 9 = 14 \
a_3 &= a_2 + d = 14 + 9 = 23
\end{align*}
$$

**Answer:** The first three terms are **5, 14, 23**.

---

### **Part (b)**
**Given:**
- 5th term ($$ a_5 $$) = 19
- 15th term ($$ a_{15} $$) = 59

**Find:** The first three terms of the sequence.

**Solution:**

1. **Set Up Equations:**

$$
\begin{cases}
a_5 = a_1 + 4d = 19 \quad \text{(3)}\
a_{15} = a_1 + 14d = 59 \quad \text{(4)}
\end{cases}
$$

2. **Subtract Equation (3) from Equation (4):**

$$
(a_1 + 14d) - (a_1 + 4d) = 59 - 19 \
10d = 40 \
d = \frac{40}{10} \
d = 4
$$

3. **Find $$ a_1 $$ Using Equation (3):**

$$
a_1 + 4 \cdot 4 = 19 \
a_1 + 16 = 19 \
a_1 = 19 - 16 \
a_1 = 3
$$

4. **Determine the First Three Terms:**

$$
\begin{align*}
a_1 &= 3 \
a_2 &= a_1 + d = 3 + 4 = 7 \
a_3 &= a_2 + d = 7 + 4 = 11
\end{align*}
$$

**Answer:** The first three terms are **3, 7, 11**.

---

### **Additional Problem**
**Given:**
- 15th term ($$ a_{15} $$) = 100
- 3rd term ($$ a_3 $$) = 28

**Find:** The 100th term ($$ a_{100} $$).

**Solution:**

1. **Set Up Equations:**

$$
\begin{cases}
a_{15} = a_1 + 14d = 100 \quad \text{(5)}\
a_3 = a_1 + 2d = 28 \quad \text{(6)}
\end{cases}
$$

2. **Subtract Equation (6) from Equation (5):**

$$
(a_1 + 14d) - (a_1 + 2d) = 100 - 28 \
12d = 72 \
d = \frac{72}{12} \
d = 6
$$

3. **Find $$ a_1 $$ Using Equation (6):**

$$
a_1 + 2 \cdot 6 = 28 \
a_1 + 12 = 28 \
a_1 = 28 - 12 \
a_1 = 16
$$

4. **Find the 100th Term:**

$$
a_{100} = a_1 + 99d = 16 + 99 \cdot 6 = 16 + 594 = 610
$$

**Answer:** The 100th term is **610**.

---
**Part (a):** - First three terms: 5, 14, 23 **Part (b):** - First three terms: 3, 7, 11 **Additional Problem:** - 100th term: 610

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