Question 6 For the series $$ 27,-9,3,-1 \ldots $$, find: (i) the first term and the common ratio (ii) the sum of the first 6 terms

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Let's analyze the given geometric series and solve each part step-by-step.

**Given Series:**
$$ 27, -9, 3, -1, \ldots $$

### (i) First Term and Common Ratio

**First Term ($$ a $$):**
The first term of the series is clearly:
$$ a = 27 $$

**Common Ratio ($$ r $$):**
The common ratio is found by dividing the second term by the first term:
$$ r = \frac{\text{Second term}}{\text{First term}} = \frac{-9}{27} = -\frac{1}{3} $$

**Verification:**
To ensure that this ratio is consistent:
- $$ -9 \times \left(-\frac{1}{3}\right) = 3 $$
- $$ 3 \times \left(-\frac{1}{3}\right) = -1 $$

This confirms that the ratio $$ r = -\frac{1}{3} $$ is consistent throughout the series.

### (ii) Sum of the First 6 Terms

To find the sum of the first $$ n $$ terms of a geometric series, we use the formula:
$$ S_n = a \frac{1 - r^n}{1 - r} $$
Where:
- $$ a = 27 $$
- $$ r = -\frac{1}{3} $$
- $$ n = 6 $$

**Plugging in the values:**
$$ S_6 = 27 \times \frac{1 - \left(-\frac{1}{3}\right)^6}{1 - \left(-\frac{1}{3}\right)} $$

**Calculating $$ r^6 $$:**
$$ \left(-\frac{1}{3}\right)^6 = \left(\frac{1}{3}\right)^6 = \frac{1}{729} $$

**Substituting back:**
$$ S_6 = 27 \times \frac{1 - \frac{1}{729}}{1 + \frac{1}{3}} = 27 \times \frac{\frac{728}{729}}{\frac{4}{3}} $$

**Simplifying the fraction:**
$$ S_6 = 27 \times \frac{728}{729} \times \frac{3}{4} $$
$$ S_6 = 27 \times \frac{728 \times 3}{729 \times 4} $$
$$ S_6 = 27 \times \frac{2184}{2916} $$
$$ S_6 = 27 \times \frac{182}{243} $$
$$ S_6 = \frac{27 \times 182}{243} = \frac{182}{9} $$

**Final Answer:**
$$ S_6 = \frac{182}{9} \approx 20.\overline{2} $$

### **Summary**

1. **First Term ($$ a $$)**: $$ 27 $$
2. **Common Ratio ($$ r $$)**: $$ -\frac{1}{3} $$
3. **Sum of the First 6 Terms ($$ S_6 $$)**: $$ \frac{182}{9} $$ or approximately $$ 20.\overline{2} $$
First term: 27 Common ratio: -1/3 Sum of the first 6 terms: 182/9 or approximately 20.2

Question 6 For the series \( 27,-9,3,-1 \ldots \), find: (i) the first term and the common ratio (ii) the sum of the first 6 terms

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