The first term in a geometric sequence is 54, and the 5th term is 2 / 3. Find an explicit form for the geometric sequence.

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=545th 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]