This problem seems to involve a geometric progression (GP), where we are given specific terms and asked to find the first term, the common ratio, and the number of terms that sum to a particular value.

(i) Find the first term and the common ratio

In a geometric progression, each term can be expressed as:

$a_n = a \cdot r^{n-1}$

where:

• $a_n$ is the $n$-th term,
• $a$ is the first term,
• $r$ is the common ratio, and
• $n$ is the term number.

Given:

• The second term ($a_2$) is $-6$,
• The fifth term ($a_5$) is $162$.

We can write the following equations:

$a_2 = a \cdot r^1 = -6$ $a_5 = a \cdot r^4 = 162$

Step 1: Solve for the ratio $r$

Divide the second equation by the first to eliminate $a$:

$\frac{a \cdot r^4}{a \cdot r} = \frac{162}{-6}$

This simplifies to:

$r^3 = -27$

Taking the cube root of both sides:

$r = \sqrt[3]{-27} = -3$

Step 2: Solve for the first term $a$

Using the value of $r$ in the equation for the second term:

$a \cdot (-3) = -6$

$a = \frac{-6}{-3} = 2$

So, the first term $a$ is 2, and the common ratio $r$ is $-3$.

(ii) Find the number of terms $n$ for which the sum is 1094

The sum $S_n$ of the first $n$ terms of a geometric series is given by:

S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{(for r \neq 1)}

Given $S_n = 1094$, $a = 2$, and $r = -3$, we substitute these into the formula:

$1094 = 2 \cdot \frac{1 - (-3)^n}{1 - (-3)}$

Simplifying:

$1094 = 2 \cdot \frac{1 - (-3)^n}{4}$

Multiply both sides by 4:

$4376 = 2 \cdot \left(1 - (-3)^n\right)$

Divide by 2:

$2188 = 1 - (-3)^n$

Subtract 1 from both sides:

$2187 = (-3)^n$

Now, express $2187$ as a power of $-3$:

$2187 = 3^7$

Since $(-3)^n = 3^7$, it follows that:

$n = 7$

So, the number of terms $n$ is 7.

Summary of Answers:

1. First Term $a$: $2$
2. Common Ratio $r$: $-3$
3. Number of Terms $n$: $7$

Would you like a more detailed explanation on any part of this problem?

Here are some related questions you might find useful:

1. How do you determine the nth term in a geometric progression?
2. What are the properties of a geometric series when the common ratio is negative?
3. Can you derive the formula for the sum of the first n terms of a geometric series?
4. How does changing the common ratio affect the sum of a geometric series?
5. What happens to the sum if the common ratio $r$ equals 1?

Tip: Always check if the common ratio is positive or negative as it greatly influences the behavior of the sequence.