Problem #1
The “rule of 72” is an approximation used to determine the number of years it will take for an investment to double given a certain annual percentage rate (APR). For example, at 6% APR, an investment will double in approximately 72 / 6 = 12 years.
a) Using your favorite computer package and the bank model we developed in class, numerically determine how many months it would take for an initial investment to double at 6% APR.
b) Solve the difference equation y[k] in closed form, assuming y[0] is the initial investment.
c) Using the Taylor series approximation:

In (1+ β) = β -½β² + ⅓β³ - ...

Use the answer from part b) and the first term of the Taylor Series expansion. Solve for how many months it would take for an initial investment to double. Compare the answer to a) .
d) Convert this formula into years
e) We use 72 instead of the number you obtained in d). Why?

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