QUESTION 2 Given the arithmetic scquence: $$ w-2 ; 2 w-4 ; 23-w $$ 2.1.1 Determine the of $$ w $$. 2.1.2 Write down the common difference. 2.2 Given the geometric sequence: $$ -\frac{1}{4} ; b ;-1 ; \ldots $$. $$ \begin{array}{ll} ext { 2.2.1 } & ext { Calculate the possible values of } b ext {. } \ ext { 2.2.2 } & ext { If } b=\frac{1}{2} ext {, calculate the } 19^{i l} ext { term }\left(T_{19} ight) ext { of the sequence. }\end{array} $$

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QUESTION 2 Given the arithmetic scquence: $$ w-2 ; 2 w-4 ; 23-w $$ 2.1.1 Determine the of $$ w $$. 2.1.2 Write down the common difference. 2.2 Given the geometric sequence: $$ -\frac{1}{4} ; b ;-1 ; \ldots $$. $$ \begin{array}{ll}	ext { 2.2.1 } & 	ext { Calculate the possible values of } b 	ext {. } \ 	ext { 2.2.2 } & 	ext { If } b=\frac{1}{2} 	ext {, calculate the } 19^{i l} 	ext { term }\left(T_{19}ight) 	ext { of the sequence. }\end{array} $$

UpStudy AI · Accepted Answer

**Arithmetic Sequence**

1. We are given the sequence:
   $$
   w-2,\quad 2w-4,\quad 23-w
   $$
   For an arithmetic sequence, the difference between consecutive terms is constant. So,
   $$
   (2w-4) - (w-2) = (23-w) - (2w-4)
   $$

2. Compute the left-hand side (LHS):
   $$
   (2w-4) - (w-2) = 2w-4-w+2 = w-2.
   $$

3. Compute the right-hand side (RHS):
   $$
   (23-w) - (2w-4) = 23-w-2w+4 = 27-3w.
   $$

4. Set the LHS equal to the RHS:
   $$
   w-2 = 27-3w
   $$

5. Solve for $$w$$:
   - Add $$3w$$ to both sides:
     $$
     4w-2 = 27
     $$
   - Add $$2$$ to both sides:
     $$
     4w = 29
     $$
   - Divide both sides by $$4$$:
     $$
     w = \frac{29}{4}
     $$

6. The common difference $$d$$ of the arithmetic sequence is:
   $$
   d = (2w-4) - (w-2) = w-2.
   $$
   Substitute $$w = \frac{29}{4}$$:
   $$
   d = \frac{29}{4} - 2 = \frac{29}{4} - \frac{8}{4} = \frac{21}{4}.
   $$

---

**Geometric Sequence**

We are given the sequence:
$$
-\frac{1}{4},\quad b,\quad -1,\ \ldots
$$
Let the common ratio be $$r$$.

1. The relationship between the terms is:
   $$
   b = -\frac{1}{4} \cdot r
   $$
   and
   $$
   -1 = b \cdot r.
   $$

2. Substitute the expression for $$b$$ into the second equation:
   $$
   -1 = \left(-\frac{1}{4} \cdot r\right) \cdot r = -\frac{1}{4} \cdot r^2.
   $$

3. Solve for $$r^2$$:
   $$
   -1 = -\frac{1}{4} \cdot r^2 \quad \Longrightarrow \quad 1 = \frac{1}{4} \cdot r^2,
   $$
   $$
   r^2 = 4.
   $$
   Hence,
   $$
   r = 2 \quad \text{or} \quad r = -2.
   $$

4. Now, find the possible values of $$b$$:
   - If $$r = 2$$:
     $$
     b = -\frac{1}{4} \cdot 2 = -\frac{1}{2}.
     $$
   - If $$r = -2$$:
     $$
     b = -\frac{1}{4} \cdot (-2) = \frac{1}{2}.
     $$
   
   Thus, the possible values of $$b$$ are $$ -\frac{1}{2} $$ and $$ \frac{1}{2} $$.

5. It is given that $$ b = \frac{1}{2} $$. From our work, this corresponds to $$ r = -2 $$.

6. The $$19^{\text{th}}$$ term $$ T_{19} $$ of a geometric sequence is given by:
   $$
   T_{19} = a \cdot r^{18}
   $$
   where $$ a = -\frac{1}{4} $$.

Substitute the values:
   $$
   T_{19} = -\frac{1}{4} \cdot (-2)^{18}.
   $$
   Since $$(-2)^{18} = 2^{18}$$ (because the exponent is even), we have:
   $$
   T_{19} = -\frac{1}{4} \cdot 2^{18}.
   $$
   Note that $$ 2^{18} = 262144 $$, thus:
   $$
   T_{19} = -\frac{1}{4} \times 262144 = -65536.
   $$

---

**Summary of Answers**

- $$w = \frac{29}{4}$$.
- Common difference of the arithmetic sequence: $$\frac{21}{4}$$.
- Possible values of $$b$$: $$-\frac{1}{2}$$ and $$\frac{1}{2}$$.
- If $$b=\frac{1}{2}$$, then $$T_{19} = -65536$$.
**Arithmetic Sequence:** - $$ w = \frac{29}{4} $$ - Common difference: $$ \frac{21}{4} $$ **Geometric Sequence:** - Possible values of $$ b $$: $$ -\frac{1}{2} $$ and $$ \frac{1}{2} $$ - If $$ b = \frac{1}{2} $$, then the 19th term $$ T_{19} = -65536 $$

QUESTION 2
Given the arithmetic scquence:
2.1.1 Determine the of .
2.1.2 Write down the common difference.
2.2 Given the geometric sequence: .

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QUESTION 2 Given the arithmetic scquence: 2.1.1 Determine the of . 2.1.2 Write down the common difference. 2.2 Given the geometric sequence: .