Answer:
The recursive formulas is [tex]a_n=3\cdot a_{n-1}[/tex].
Step-by-step explanation:
The geometric sequence is defined by the formula
[tex]a_n=2\cdot3^{n-1}[/tex]
The first term of the sequence is
[tex]a_1=2\cdot3^{1-1}=2\cdot3^0=2[/tex]
The second term of the sequence is
[tex]a_2=2\cdot3^{2-1}=2\cdot3=6[/tex]
Similarly (n-1)th term is
[tex]a_{n-1}=2\cdot3^{n-1-1}=2\cdot3^{n-2}[/tex]
The common ratio of a geometric sequence is
[tex]r=\frac{a_2}{a_1}=\frac{6}{2}=3[/tex]
The recursive formulas for the geometric sequence is calculated as
[tex]a_n=2\cdot3^{n-1}[/tex]
[tex]a_n=2\cdot3^{n-1-1}\cdot 3[/tex] [tex][\because a^{m+n}=a^ma^n][/tex]
[tex]a_n=2\cdot3^{n-2}\cdot 3[/tex]
[tex]a_n=a_{n-1}\cdot 3[/tex] [tex][\because a_{n-1}=2\cdot3^{n-2}][/tex]
[tex]a_n=3\cdot a_{n-1}[/tex]
Therefore the recursive formulas is [tex]a_n=3\cdot a_{n-1}[/tex].
