Answer:
Recursive formula for geometric sequence
[tex]a_{n}=a_{n-1}\times r[/tex]
is [tex]a_{n}=a_{n-1}\times 6[/tex]
and explicit formula for geometric sequence [tex]a_{n}=a_{1}^{r-1}[/tex] is
[tex]a_{n}=(\frac{1}{2})^{6-1}[/tex]
Step-by-step explanation:
Given sequence is [tex]\frac{1}{2},3,18,108,648,...[/tex]
To find the recursive and explicit formula for this sequence:
Let [tex]a_{1}=\frac{1}{2},a_{2}=3,a_{3}=18,a_{4}=108,a_{5}=648,...[/tex]
To find the common ratio r:
[tex]r=\frac{a_{2}}{a_{1}}[/tex]
[tex]=\frac{3}{\frac{1}{2}}[/tex]
[tex]=3\times 2[/tex]
[tex]=6[/tex]
Therefore r=6
[tex]r=\frac{a_{3}}{a_{2}}[/tex]
[tex]=\frac{18}{3}[/tex]
[tex]=6[/tex]
Therefore r=6
Therefore the common ration r=6
Therefore the given sequence is geometric sequence
Recursive formula for geometric sequence is [tex]a_{n}=a_{n-1}\times r[/tex]
[tex]a_{n}=a_{n-1}\times 6[/tex]
and explicit formula is [tex]a_{n}=a_{1}^{r-1}[/tex]
[tex]a_{n}=(\frac{1}{2})^{6-1}[/tex]
[tex]=(\frac{1}{2})^{5}[/tex]
[tex]=\frac{1}{32}[/tex]
Therefore [tex]a_{n}=\frac{1}{32}[/tex]
