To determine the parameters necessary for using the given geometric series formula [tex]\( S_x = \frac{r t_x - t_1}{r - 1}, \quad r \neq 1 \)[/tex], let's examine what each component of the formula represents and what we need to know to use it effectively:
1. [tex]\( r \)[/tex]: This is the common ratio of the geometric series.
2. [tex]\( t_x \)[/tex]: This term represents the last term of the geometric progression (GP).
3. [tex]\( t_1 \)[/tex]: This is the first term of the geometric series.
4. [tex]\( x \)[/tex] (implicitly from [tex]\( t_x \)[/tex]): This is the total number of terms in the geometric series.
The formula [tex]\( S_x = \frac{r t_x - t_1}{r - 1} \)[/tex], for [tex]\( r \neq 1 \)[/tex], calculates the sum of the first [tex]\( x \)[/tex] terms of a geometric series. To use this formula, we need:
- The common ratio [tex]\( r \)[/tex]
- The first term [tex]\( t_1 \)[/tex]
- The total number of terms [tex]\( x \)[/tex]
The last term [tex]\( t_x \)[/tex] can be derived if we know the total number of terms, the first term, and the common ratio.
Therefore, the combination of parameters required to use this formula is the common ratio [tex]\( r \)[/tex], the first term [tex]\( t_1 \)[/tex], and the total number of terms [tex]\( x \)[/tex].
Thus, the correct choice is:
- the common ratio, the first term, and the total number of terms
