In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio $r$. Given the first four terms of a geometric sequence, the equation for the $n$-th term $a_n$ is:

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

Where:

• $a_n$ is the $n$-th term.
• $a_1$ is the first term of the sequence.
• $r$ is the common ratio.
• $n$ is the term number.

Steps to derive the equation:

1. Find the common ratio:

   • From the first four terms $a_1, a_2, a_3, a_4$, the common ratio $r$ can be determined as: $r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3}$

2. Write the general formula:

   • Once you have the first term $a_1$ and the common ratio $r$, you can substitute these values into the general formula: $a_n = a_1 \cdot r^{(n-1)}$

Would you like to work through an example with specific values for the first four terms, or do you have any clarifications?

Related questions:

1. How can you find the common ratio if only the second and third terms are given?
2. How does the geometric sequence differ from an arithmetic sequence?
3. What happens to the sequence when the common ratio is less than 1?
4. Can the common ratio be negative, and what effect does it have on the sequence?
5. How do you sum the first $n$ terms of a geometric sequence?

Tip:

When working with geometric sequences, always check the ratio between consecutive terms to ensure it's constant throughout the sequence.