To find the recursive formula of a geometric sequence when given its explicit formula, we need to express each term [tex]\(a_n\)[/tex] in terms of the previous term [tex]\(a_{n-1}\)[/tex].
Given the explicit formula:
[tex]\[ a_n = 13 \cdot (-2)^{n-1} \][/tex]
We want to express [tex]\(a_n\)[/tex] in terms of [tex]\(a_{n-1}\)[/tex]. To do this, let's restate the formula for [tex]\(a_{n-1}\)[/tex]:
[tex]\[ a_{n-1} = 13 \cdot (-2)^{(n-1)-1} = 13 \cdot (-2)^{n-2} \][/tex]
Now, we recognize that [tex]\(a_n\)[/tex] can be written as:
[tex]\[ a_n = 13 \cdot (-2)^{n-1} \][/tex]
To find the relationship between [tex]\(a_n\)[/tex] and [tex]\(a_{n-1}\)[/tex], consider the ratio:
[tex]\[ \frac{a_n}{a_{n-1}} = \frac{13 \cdot (-2)^{n-1}}{13 \cdot (-2)^{n-2}} \][/tex]
By simplifying this ratio, we get:
[tex]\[ \frac{a_n}{a_{n-1}} = \frac{(-2)^{n-1}}{(-2)^{n-2}} \][/tex]
Since [tex]\((-2)^{n-1} = (-2) \cdot (-2)^{n-2}\)[/tex], we can further simplify the fraction:
[tex]\[ \frac{a_n}{a_{n-1}} = (-2) \][/tex]
Therefore, the common ratio [tex]\(r\)[/tex] is [tex]\(-2\)[/tex]. Using this, we can write the recursive formula as:
[tex]\[ a_n = -2 \cdot a_{n-1} \][/tex]
Thus, the recursive formula for the given geometric sequence is:
[tex]\[ \boxed{a_n = -2 \cdot a_{n-1}} \][/tex]
