find the 1st term and the common ratio of the given date : a4=-12 and a5=-6

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!

Learn how to find the first term and common ratio of a geometric sequence using given terms a4=-12 and a5=-6. This problem involves using the formula for the n-th term of a geometric sequence and calculating ratios between consecutive terms. Suitable for high school students studying geometric sequences.

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