Answer:
Explicit formula will be [tex]A_n=3^{(2-n)}[/tex]
Step-by-step explanation:
Explicit formula for a geometric sequence is given by the formula,
[tex]A_n=a(r)^{n-1}[/tex]
Here [tex]A_n[/tex] is the nth term of the sequence
a = first term of the sequence
n = number of term
Since 5th term of a geometric sequence is
[tex]A_5[/tex] = [tex]\frac{1}{27}[/tex]
First term of the sequence 'a'= [tex]\frac{1}{3}[/tex]
By substituting these values in the explicit formula,
[tex]\frac{1}{27}=a(\frac{1}{3})^{5-1}[/tex]
[tex]\frac{1}{27} =a(\frac{1}{3})^4[/tex]
a = [tex]\frac{\frac{1}{27}}{\frac{1}{81}}[/tex]
a = [tex]\frac{81}{27}[/tex]
a = 3
Therefore, explicit formula of this sequence will be,
[tex]A_n=(3)(\frac{1}{3})^{n-1}[/tex]
[tex]A_n=(3)(3)^{(-n+1)}[/tex]
[tex]A_n=3^{(2-n)}[/tex]
