Answer:
[tex]1296[/tex] would be the term immediately after the term [tex]216[/tex].
Step-by-step explanation:
The recursive formula for the [tex]n[/tex]th term a geometric series is:
[tex]a_{n} = r\, a_{n - 1}[/tex],
Where:
- [tex]r[/tex] is the common ratio of this series, and
- [tex]a_{n - 1}[/tex] denotes the [tex](n-1)[/tex]th term of the series, which is the term immediately before the [tex]n[/tex]th term.
The common ratio of a geometric series is the ratio between each term and the term before it- for example, the ratio between the term [tex]36[/tex] and the term right before it, [tex]6[/tex]. Using this property, the common ratio of the geometric series in this question would be:
[tex]\displaystyle r = \frac{36}{6} = 6[/tex].
In this question, if [tex]a_{n}[/tex] represents the unknown term, [tex]216[/tex] would be the value of [tex]a_{n-1}[/tex], which is immediately before that unknown term. Since the common ratio is [tex]r = 6[/tex], the value of the unknown term [tex]a_{n}[/tex] can be found using the recursive formula as follows:
[tex]a_{n} = r\, a_{n - 1} = (6) \times (216) = 1296[/tex].
