The second term of a positive magnitude is -6 and the fifth term is 162. (i)

Find the first term of the magnitude and the common ratio.

(ii) The sum of a certain number of terms from the first term of this scale is 1094. Find the number of terms.

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.

Explore how to solve geometric progression problems with specific examples. Learn to find the first term and common ratio from given terms and calculate the number of terms to sum to a specific value. Detailed step-by-step solutions included.

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