To find the correct recursive definition for the sequence given as [tex]\(7, 21, 63, 189\)[/tex], we will look at how each term in the sequence relates to its predecessor.
Given the sequence:
- [tex]\(a_1 = 7\)[/tex]
- [tex]\(a_2 = 21\)[/tex]
- [tex]\(a_3 = 63\)[/tex]
- [tex]\(a_4 = 189\)[/tex]
We want to identify the pattern.
First, let's determine the ratios between consecutive terms:
- [tex]\( \frac{a_2}{a_1} = \frac{21}{7} = 3 \)[/tex]
- [tex]\( \frac{a_3}{a_2} = \frac{63}{21} = 3 \)[/tex]
- [tex]\( \frac{a_4}{a_3} = \frac{189}{63} = 3 \)[/tex]
The ratios tell us that each term is three times the preceding term. Therefore, the pattern [tex]\(a_n = 3 \times a_{n-1}\)[/tex] can be noted.
To explicitly define this pattern:
- The first term [tex]\(a_1 = 7\)[/tex]
- The [tex]\(n\)[/tex]-th term is obtained by multiplying the [tex]\((n-1)\)[/tex]-th term by 3.
Thus, the recursive definition of the sequence is:
[tex]\[
\left\{\begin{array}{l}
a_n = 3 a_{n-1} \\
a_1 = 7
\end{array}\right.
\][/tex]
This definition accurately represents the relationship between the terms provided in the sequence [tex]\(7, 21, 63, 189\)[/tex].
