91Ó°ÊÓ

In any geometric sequence, the first term is crucial as it serves as the starting point of the sequence. The first term is commonly represented by the letter \(a\). It sets the basis for the entire sequence. For the given sequence \(-2, -6, -18, \,\dots\), the first term is clearly \(-2\). This means that \(a = -2\). It's important to identify this first term correctly since it is used in the general term formula later.

The common ratio in a geometric sequence determines how the sequence progresses. It shows how we move from one term to the next. The common ratio, denoted by \(r\), is found by dividing any term by the previous term in the sequence. For our sequence, we can calculate this by taking the second term and dividing it by the first term:
- Second term: \(-6\)
- First term: \(-2\)
This results in:
\( r = \frac{-6}{-2} = 3 \).
Hence, the common ratio, \(r\), is \(3\). This ratio should be constant between all terms of the sequence, ensuring that the sequence follows a geometric progression.

The general term formula in a geometric sequence allows us to find any term in the sequence without having to list all the terms one by one. This formula is particularly handy for sequences with a large number of terms. The general term is given by:
\( a_n = a \times r^{(n-1)} \).
Here, \(a_n\) represents the \(n\)-th term of the sequence, \(a\) is the first term, and \(r\) is the common ratio. Substituting the known values from our sequence, where \(a = -2\) and \(r = 3\), we get:
\( a_n = -2 \times 3^{(n-1)} \).
This formula allows you to find any term in the sequence efficiently. For instance, to find the 4th term, substitute \(n = 4\) into the formula:
\( a_4 = -2 \times 3^{(4-1)} = -2 \times 27 = -54\).