Answer:
a_n = 39(1.06)^(n - 1)
Step-by-step explanation:
We are told that a_n is the number of raccoons in a given n number of years.
After 6 years, there are 52 raccoons.
Also after 8 years there are 58 raccoons.
Using formula for geometric sequence, we have;
a_n = ar^(n - 1)
After 6 years, there are 52 raccoons
Thus;
a_6 = 52 = ar^(6 - 1)
ar^(5) = 52 - - - (eq 1)
Also after 8 years there are 58 raccoons. Thus;
a_8 = 58 = ar^(8 - 1)
ar^(7) = 58 - - - (eq 2)
Divide eq 1 by eq 2 to get;
(r^(7))/(r^(5)) = 58/52
r² = 58/52
r = √(58/52)
r ≈ 1.06
Put 1.06 for r in eq 1 to get;
a(1.06)^(5) = 52
1.3382a = 52
a = 52/1.3382
a ≈ 39
Thus,the explicit rule is;
a_n = 39(1.06)^(n - 1)
