3.1 The $$ 4^{ ext {th }} $$ term of an arithmetic sequence is -3 and the $$ 20^{t h} $$ term is -35 . Determine the common difference and the first term. 3.2 Evaluate : $$ \sum_{k=1}^{20} 3^{k-2} $$ 3.3 The following sequence forms a convergent geometric sequence: $$ \frac{3}{(x-1)^{2}}+\frac{1}{(x-1)}+\frac{1}{3}+\frac{(x-1)}{9}+\ldots $$ $$ 3.3 .1 \quad $$ Determine the possible values of x . 3.3 .2 If $$ x=2 $$, calculate $$ S_{\infty} $$.

Question

3.1 The $$ 4^{	ext {th }} $$ term of an arithmetic sequence is -3 and the $$ 20^{t h} $$ term is -35 . Determine the common difference and the first term. 3.2 Evaluate : $$ \sum_{k=1}^{20} 3^{k-2} $$ 3.3 The following sequence forms a convergent geometric sequence: $$ \frac{3}{(x-1)^{2}}+\frac{1}{(x-1)}+\frac{1}{3}+\frac{(x-1)}{9}+\ldots $$ $$ 3.3 .1 \quad $$ Determine the possible values of x . 3.3 .2 If $$ x=2 $$, calculate $$ S_{\infty} $$.

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### **3.1 Arithmetic Sequence**

**Given:**
- The $$4^{\text{th}}$$ term ($$a_4$$) is $$-3$$.
- The $$20^{\text{th}}$$ term ($$a_{20}$$) is $$-35$$.

**In an arithmetic sequence:**
$$
a_n = a + (n-1)d
$$
where:
- $$a$$ is the first term.
- $$d$$ is the common difference.

**Setting up the equations:**
$$
\begin{cases}
a + 3d = -3 \quad \text{(for } a_4) \
a + 19d = -35 \quad \text{(for } a_{20})
\end{cases}
$$

**Subtract the first equation from the second:**
$$
(a + 19d) - (a + 3d) = -35 - (-3) \
16d = -32 \
d = -2
$$

**Find the first term ($$a$$):**
$$
a + 3(-2) = -3 \
a - 6 = -3 \
a = 3
$$

**Answer:**
- **First term ($$a$$)**: $$3$$
- **Common difference ($$d$$)**: $$-2$$

---

### **3.2 Sum of a Geometric Series**

**Given:**
$$
\sum_{k=1}^{20} 3^{k-2}
$$

**Simplify the expression:**
$$
3^{k-2} = \frac{3^k}{9}
$$
Thus, the sum becomes:
$$
S = \frac{1}{9} \sum_{k=1}^{20} 3^k = \frac{1}{9} \cdot \sum_{k=1}^{20} 3^k
$$

**Recognize it as a geometric series:**
- **First term ($$a$$)**: $$3^{-2} = \frac{1}{9}$$
- **Common ratio ($$r$$)**: $$3$$
- **Number of terms ($$n$$)**: $$20$$

**Sum of a geometric series formula:**
$$
S = a \cdot \frac{r^n - 1}{r - 1}
$$
$$
S = \frac{1}{9} \cdot \frac{3^{20} - 1}{3 - 1} = \frac{3^{20} - 1}{18}
$$

**Answer:**
$$
\sum_{k=1}^{20} 3^{k-2} = \frac{3^{20} - 1}{18}
$$

---

### **3.3 Geometric Sequence Analysis**

#### **3.3.1 Determine Possible Values of $$x$$**

**Given Sequence:**
$$
\frac{3}{(x-1)^2},\ \frac{1}{(x-1)},\ \frac{1}{3},\ \frac{(x-1)}{9},\ \ldots
$$

**Identify the common ratio ($$r$$):**
$$
r = \frac{\text{Second term}}{\text{First term}} = \frac{\frac{1}{x-1}}{\frac{3}{(x-1)^2}} = \frac{(x-1)}{3}
$$

**For the sequence to be convergent:**
$$
|r| < 1 \Rightarrow \left|\frac{x-1}{3}\right| < 1
$$
$$
|x - 1| < 3 \Rightarrow -3 < x - 1 < 3 \Rightarrow -2 < x < 4
$$

**Answer:**
$$
x \in (-2,\, 4)
$$

#### **3.3.2 Calculate $$ S_{\infty} $$ When $$x = 2$$**

**Given $$x = 2$$:**
$$
r = \frac{2 - 1}{3} = \frac{1}{3}
$$

**First term ($$a$$):**
$$
a = \frac{3}{(2-1)^2} = 3
$$

**Sum of an infinite geometric series formula (since $$|r| < 1$$):**
$$
S_{\infty} = \frac{a}{1 - r} = \frac{3}{1 - \frac{1}{3}} = \frac{3}{\frac{2}{3}} = \frac{9}{2}
$$

**Answer:**
$$
S_{\infty} = \frac{9}{2}
$$

---

### **Summary of Answers:**

- **3.1**
  - First term ($$a$$): **3**
  - Common difference ($$d$$): **$$-2$$**
  
- **3.2**
  $$
  \sum_{k=1}^{20} 3^{k-2} = \frac{3^{20} - 1}{18}
  $$
  
- **3.3.1**
  $$
  x \in (-2,\, 4)
  $$
  
- **3.3.2**
  $$
  S_{\infty} = \frac{9}{2}
  $$
- **3.1** - First term: **3** - Common difference: **-2** - **3.2** $$ \sum_{k=1}^{20} 3^{k-2} = \frac{3^{20} - 1}{18} $$ - **3.3.1** $$ x \in (-2,\, 4) $$ - **3.3.2** $$ S_{\infty} = \frac{9}{2} $$

3.1 The term of an arithmetic sequence is -3 and the term is -35 .
Determine the common difference and the first term.
3.2 Evaluate :
3.3 The following sequence forms a convergent geometric sequence:

Determine the possible values of x .
3.3 .2 If , calculate .

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3.1 The term of an arithmetic sequence is -3 and the term is -35 . Determine the common difference and the first term. 3.2 Evaluate : 3.3 The following sequence forms a convergent geometric sequence: Determine the possible values of x . 3.3 .2 If , calculate .