mean challenge..

You can simplify the progression like this

3^2, a, b, 3^5

So clearly a = 3^3 = 27 and b = 3^4 = 81

Arithmetic mean (or average) is (27 +81)/2 = 54

By the property of the geometric progression, we get a.b=9.243=32.35=37. We also know that a9=ba, or a29=b Substituting into the previous equation, we get a39=37, or a3=39, which means that a=33=27. Then b=a29=81 and their arithmetic mean is 27+812=1082=54.

Write a solution. Since the problem is a geometric progression, with a given first term of 9, I began by finding the common ratio. The fourth term is 243. Mathematically, one can say that the fourth term is equal to the first term multiplied by the common ratio raised to the third power. In general:

The nth term = first term * common ratio ^ (n-1)

Substitute and solve for r:

243 = 9r^3

27 = r^3

3 = r

To get a and b, multiply by the common ratio.

a = 9*3 = 27

b = 27*3 = 81

The arithmetic mean is just the sum of these numbers, divided by 2.

(27+81)/2 = 108/2 = 54