gmatter2012 wrote:Ax(y) is an operation that adds 1 to y and then multiplies the result by x. If x = 2/3, then Ax(Ax(Ax(Ax(Ax(x))))) is between
Algebraic Approach:
Ax(y) = (y + 1)*x
Ax(x) = (x + 1)*x = (x^2 + x)
Ax(Ax(x)) = Ax(x^2 + x) = (x^2 + x + 1)*x = (x^3 + x^2 + x)
Ax(Ax(Ax(x))) = Ax(x^3 + x^2 + x) = (x^3 + x^2 + x + 1)*x = (x^4 + x^3 + x^2 + x)
...
Ax(Ax(Ax(Ax(Ax(x))))) = (x^6 + x^5 + x^4 + x^3 + x^2 + x)
This is a geometric series with first term = x and common ratio = x.
Hence, the sum = x(1 - x^6)/(1 - x)
For x = 2/3, the sum = (2/3)*(1 - (2/3)^6)/(1 - 2/3) = (2/3)*((1 - (2/3)^6)/(1/3) = 2*(1 - (2/3)^6)
Now, (2/3)^6 will be a very small number with respect to 1, the sum will be smaller than 2 but very close to it.
The correct answer is D.
