Question: what term is the number .006561 in the sequence 30, 9, 2.7 ?
The sequence is a G.P., with initial term a1 = 30 and ratio of proportionality r = 0.3.
The nth term is given by: an = a1*r^(n-1)
When an = 0.006561, then
a1*r^(n-1) = an = 0.006561
30*(0.3)^(n-1) = 0.006561
(0.3)^(n-1) = 2.187*10^(-4)
taking logs of both sides,
(n-1)ln(0.3) = ln(2.187*10^(-4))
n-1 = ln(2.187*10^(-4)) / ln(0.3) = -8.442781 / -1.20397 = 7
n-1 = 7
n = 8
Answer: 0.006561 is the 8th term of the sequence