If the second and the fifth terms of a geometric sequence. Respectively, are 20%9 and 160%243,then find A, the common ratio? B,the first term ? C,the (n+1)term ?

Let's denote the first term of the geometric sequence as "a" and the common ratio as "r".

We are given that the second term is 20%9 and the fifth term is 160%243.

The second term can be written as a*r, and we have a*r = 20%9.
The fifth term can be written as a*r^4, and we have a*r^4 = 160%243.

To find the common ratio "r", we can divide the equation for the fifth term by the equation for the second term:
(a*r^4) / (a*r) = (160%243) / (20%9)
r^3 = (160%243) / (20%9)
r^3 = (160/243) / (20/9)
r^3 = (160/243) * (9/20)
r^3 = 12/27
r^3 = 4/9
r = (4/9)^(1/3)

To find the first term "a", we can substitute the value of "r" into the equation for the second term:
a*r = 20%9
a*(4/9)^(1/3) = 20%9
a = (20%9) / ((4/9)^(1/3))

To find the (n+1)th term, we can use the formula for the nth term of a geometric sequence:
a_n = a*r^(n-1)
a_(n+1) = a*r^n

So, the (n+1)th term is a*(4/9)^n.

Unfortunately, without knowing the value of "n", we cannot determine the exact value of the (n+1)th term.