We are given the following information about a geometric sequence:

• $a_4 = -12$
• $a_5 = -6$

In a geometric sequence, the general formula for the $n$-th term is:

$a_n = a_1 \cdot r^{n-1}$

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $r$ is the common ratio.

Step 1: Finding the Common Ratio ($r$)

To find the common ratio, we use the fact that the ratio of consecutive terms in a geometric sequence is constant:

$r = \frac{a_5}{a_4} = \frac{-6}{-12} = \frac{1}{2}$

Thus, the common ratio $r = \frac{1}{2}$.

Step 2: Finding the First Term ($a_1$)

Now, we know the relationship between consecutive terms in the sequence. Using $a_4 = -12$ and the formula $a_4 = a_1 \cdot r^3$, we substitute the known values:

$-12 = a_1 \cdot \left( \frac{1}{2} \right)^3$

Simplifying:

$-12 = a_1 \cdot \frac{1}{8}$

To solve for $a_1$, multiply both sides by 8:

$a_1 = -12 \times 8 = -96$

Final Answer:

• The first term $a_1 = -96$.
• The common ratio $r = \frac{1}{2}$.

Would you like more details or have any questions?

Here are five related questions:

1. How do you find the nth term in a geometric sequence given the first term and common ratio?
2. How would the problem change if the common ratio was negative?
3. What is the sum of the first $n$ terms in a geometric series?
4. How do you determine the common ratio from non-consecutive terms?
5. Can a geometric sequence have a common ratio of zero?

Tip: In a geometric sequence, the ratio between any two consecutive terms is always constant. This makes identifying patterns and solving for unknowns easier!