If you invest x dollars at 3% interest compounded annually....?

Question

If you invest x dollars at 3% interest compounded annually, then the amount A(x) of the investment after one year is A(x) = 1.03x. Find each of the following.

A(A(x))

A(A(A(x)))

A(A(A(A(x))))

Find a formula for the composition of n copies of A.

Answer 1

A(A(x)) = A(1.03x) = 1.03(1.03x) = 1.03^2x

A(A(A(x))) = A(1.03^2x) = 1.03 (1.03^2x) = 1.03^3x

n copies of A is (1.03^n)x  which means 1.03 raised to the nth power times x

Answer 2

A(x) = 1.03x

A(A(x)) = 1.03 (1.03x) = 1.03^2 x

A(A(A(x))) = 1.03 (1.03^2 x) = 1.03^3 x

A(A(A(A(x)))) = 1.03 (1.03^3 x) = 1.03^4 x