determine 3rd term in the geometric sequence whose first term is -8 and whose common ratio is 6

To determine the 3rd term in a geometric sequence, you can use the formula for the \(n\)th term of a geometric sequence:

\[
a_n = a_1 \cdot r^{(n-1)}
\]

Given:
- \(a_1 = -8\)
- \(r = 6\)
- \(n = 3\)

Substitute the values into the formula:

\[
a_3 = -8 \cdot 6^{(3-1)} = -8 \cdot 6^2
\]

Calculate \(6^2\):

\[
6^2 = 36
\]

Now multiply:

\[
a_3 = -8 \cdot 36 = -288
\]

So, the 3rd term in the geometric sequence is **-288**.

Would you like further details or have any other questions?

Here are 5 related questions to explore:
1. What is the 5th term in this geometric sequence?
2. How would you find the common ratio if it wasn't given, using the first two terms?
3. What is the sum of the first 3 terms in this sequence?
4. How does the sign of the first term affect the sequence?
5. Can you derive the formula for the sum of the first \(n\) terms in this geometric sequence?

**Tip:** The sign of the first term affects the sign of the terms in the sequence. If the common ratio is positive, all terms will have the same sign as the first term.

Learn how to determine the 3rd term in a geometric sequence with a first term of -8 and a common ratio of 6 using the formula for the nth term of a geometric sequence. This problem covers key concepts in geometric sequences and involves calculating terms using specific values. Suitable for students in Grades 9-12.

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