Answer :
Answer: [tex]ar^{n-1}=54(\dfrac{1}{3})^{n-1}[/tex]
Step-by-step explanation:
The nth term for a geometric sequence is given by :-
[tex]a_n=ar^{n-1}[/tex] (1)
We are given that The first term in a geometric sequence is 54. i.e. a=54
5th term=[tex]\dfrac{2}{3}[/tex] (2)
Put n=5 and a= 54 in (1), we get
[tex]a_5=(54)r^{4}=[/tex] (3)
From (2) and (3), we have
[tex]\Rightarrow(54)r^{4}=\dfrac{2}{3}\\\\\Rightarrow r^4=\dfrac{2}{3}\times\dfrac{1}{54}=\dfrac{1}{3\times27}=\dfrac{1}{81}\\\\\Rightarrow\ r=(\dfrac{1}{81})^{\frac{1}{4}}=\dfrac{1}{3}[/tex]
Explicit form for the geometric sequence: [tex]ar^{n-1}=54(\dfrac{1}{3})^{n-1}[/tex]