QUESTION 3 3.1 The third term of a geometric series is 7 and the sixth term is -2401 . 3.1.1 Determine the first term and the common ratio. If the first terms of a geometric series are $$ 2+6+18+\cdots $$ then what is the value $$ n $$ if $$ S_{n}=177146 ? $$

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**QUESTION 3**

**3.1.1 Determine the first term and the common ratio.**

Given:
- The third term of a geometric series, $$ a_3 = 7 $$
- The sixth term of the series, $$ a_6 = -2401 $$

In a geometric series, the $$ n $$-th term is given by:
$$
a_n = a_1 \cdot r^{n-1}
$$
where:
- $$ a_1 $$ is the first term
- $$ r $$ is the common ratio

**Step 1: Express $$ a_3 $$ and $$ a_6 $$ in terms of $$ a_1 $$ and $$ r $$**
$$
a_3 = a_1 \cdot r^{2} = 7 \quad \text{(1)}
$$
$$
a_6 = a_1 \cdot r^{5} = -2401 \quad \text{(2)}
$$

**Step 2: Divide equation (2) by equation (1) to eliminate $$ a_1 $$**
$$
\frac{a_6}{a_3} = \frac{a_1 \cdot r^{5}}{a_1 \cdot r^{2}} = r^{3} = \frac{-2401}{7} = -343
$$
$$
r^{3} = -343 \quad \Rightarrow \quad r = \sqrt[3]{-343} = -7
$$

**Step 3: Substitute $$ r = -7 $$ back into equation (1) to find $$ a_1 $$**
$$
a_1 \cdot (-7)^{2} = 7
$$
$$
a_1 \cdot 49 = 7 \quad \Rightarrow \quad a_1 = \frac{7}{49} = \frac{1}{7}
$$

**Conclusion:**
- **First term ($$ a_1 $$)**: $$ \frac{1}{7} $$
- **Common ratio ($$ r $$)**: $$ -7 $$

---

**Additional Problem:**

*If the first terms of a geometric series are $$ 2 + 6 + 18 + \cdots $$, what is the value of $$ n $$ if $$ S_{n} = 177146 $$?*

**Given:**
- First term, $$ a_1 = 2 $$
- The series: $$ 2, 6, 18, \ldots $$

**Step 1: Determine the common ratio ($$ r $$)**
$$
r = \frac{6}{2} = 3
$$

**Step 2: Use the formula for the sum of the first $$ n $$ terms of a geometric series**
$$
S_n = a_1 \cdot \frac{r^{n} - 1}{r - 1}
$$
Substitute the known values:
$$
177146 = 2 \cdot \frac{3^{n} - 1}{3 - 1}
$$
$$
177146 = 2 \cdot \frac{3^{n} - 1}{2}
$$
$$
177146 = 3^{n} - 1
$$
$$
3^{n} = 177147
$$

**Step 3: Solve for $$ n $$**
Recognize that $$ 177147 $$ is a power of 3:
$$
3^{11} = 177147
$$
$$
\Rightarrow \quad n = 11
$$

**Conclusion:**
The value of $$ n $$ is **11**.
First term: $$ \frac{1}{7} $$, Common ratio: $$ -7 $$. For the series $$ 2 + 6 + 18 + \cdots $$, when $$ S_n = 177146 $$, $$ n = 11 $$.

QUESTION 3 3.1 The third term of a geometric series is 7 and the sixth term is -2401 . 3.1.1 Determine the first term and the common ratio. If the first terms of a geometric series are \( 2+6+18+\cdots \) then what is the value \( n \) if \( S_{n}=177146 ? \)

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