https://file.aiquickdraw.com/mathbot/file/1726684162987oqyvhz7g.jpg

The problem in the image is asking to find the explicit formula for the \( n \)-th term of the sequence:  
\[ 144, -24, 4, \dots \]

Let's analyze the sequence:

1. The first term (\( a_1 \)) is 144.
2. The second term (\( a_2 \)) is -24.
3. The third term (\( a_3 \)) is 4.

To find the explicit formula, let's check if this is a geometric sequence by dividing consecutive terms:
- \( \frac{-24}{144} = -\frac{1}{6} \)
- \( \frac{4}{-24} = -\frac{1}{6} \)

Since the ratio between consecutive terms is constant and equals \(-\frac{1}{6}\), the sequence is geometric with a common ratio \( r = -\frac{1}{6} \).

The explicit formula for the \( n \)-th term of a geometric sequence is:
\[ a_n = a_1 \cdot r^{n-1} \]

In this case:
- \( a_1 = 144 \)
- \( r = -\frac{1}{6} \)

Thus, the explicit formula is:
\[ a_n = 144 \cdot \left( -\frac{1}{6} \right)^{n-1} \]

Would you like further details on how geometric sequences work, or do you have any questions?

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Here are 5 related questions:
1. How do you derive the explicit formula for an arithmetic sequence?
2. What happens if the common ratio is positive instead of negative in a geometric sequence?
3. Can you calculate the 5th term of this sequence using the explicit formula?
4. How does the formula change if the initial term \( a_1 \) is negative?
5. What are the key differences between geometric and arithmetic sequences?

**Tip:** When working with sequences, always check for a pattern like common differences (arithmetic) or ratios (geometric). This helps in finding the correct formula.

Write an explicit formula for a_n, the nth term of the sequence 144, -24, 4, ...

Discover how to find the explicit formula for the nth term of the geometric sequence 144, -24, 4, ... This step-by-step solution includes key concepts of geometric sequences, suitable for students in Grades 9-10.

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