Question

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?

Answer 1

• A ○ A = 1.06²x . . . . . . . . . . the amount after 2 years
• A ○ A ○ A = 1.06³x . . . . . . the amount after 3 years
• A ○ A ○ A ○ A = 1.06⁴x . . . the amount after 4 years

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

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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) = 1.06⁴x

These are the amounts after 3 and 4 years, respectively.