81, 27, 9, 3,... Find the common ratio of the given sequence, and write an exponential function which represents the sequence. Use n = 1, 2, 3, ... A) 3; f(n) = 81^n-1 B) 3; f(n) = 81(3)^n-1 C) 1 /3 ; f(n) = 81(3)^n-1 D) 1/ 3 ; f(n) = 81(1 /3 )^n-1Answer -Since each term is multiplied by 1/3to get to the next term, the common ratio is 1/3. The common ratio is also the base of anexponential function. The correct answer is1/3; f(n) = 81(1/3)^n-1so D.

Given Sequence:
81, 27, 9 , 3 , ...
To find the common ratio:
Common ratio, r = a2/a1
r= 27/81
r=1/3

r= a3/a2= 9/27 = 1/3

r= a4/a3 = 3/9 = 1/3

So common ratio is 1/3.

Now exponential function is:
f(n) = 81 ( 1/3 )^(n-1)
When n=1
f(1)= 81 ( 1/3) ^ (1-1)
f(1)=81 ( 1/3)^0
f(1)=81(1) =81
When n=2
f(2)= 81 (1/3) ^(2-1)
f(2)= 81(1/3)^1
f(2)=27
And so on.

Answer: Option D. r= 1/3 , f(n)= 81 (1/3)^n-1