If you invest x dollars at 6% interest compounded annually, then the amount A(x) of the investment after one year is A(x) = 1.06x. Find A ○ A, A ○ A ○ A, and A ○ A ○ A ○ A. What do these compositions represent?
Accepted Solution
A:
Answer:A ○ A = 1.06²x . . . . . . . . . . the amount after 2 yearsA ○ A ○ A = 1.06³x . . . . . . the amount after 3 yearsA ○ A ○ A ○ A = 1.06⁴x . . . the amount after 4 yearsStep-by-step explanation:The composition A ○ A is equivalent to A(A(x)) which is the amount of the investment one year after the original amount matured (at one year). That is, it is the amount of the investment after 2 years. Its value is ... A(A(x)) = 1.06(1.06x) = 1.06²x__Similarly, ... A ○ A ○ A = A(A(A(x))) = 1.06(1.06(1.06x)) = 1.06³x, and A ○ A ○ A ○ A = A(A(A(A(x)))) = 1.06(1.06³x) = 1.06⁴xThese are the amounts after 3 and 4 years, respectively.