Question 368568
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The formula for the Future Value, *[tex \Large FV], of an investment *[tex \Large P] at interest rate, *[tex \Large r], percent per annum compounded *[tex \Large m] times per annum for *[tex \Large t] years is:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ FV\ =\ P\left(1\ +\ \frac{r/100}{m}\right)^{mt}]


Since you want your investment to double, *[tex \Large \frac{FV}{P}\ =\ 2], hence:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(1\ +\ \frac{2/100}{12}\right)^{12t}\ =\ 2]


And we need to solve for *[tex \Large t].  The first thing to do is use the Rule of 72 in your head to get an approximation of the answer so that you make sure your answer is at least reasonable.  Divide 72 by the interest rate to get an approximation of the time it takes an investment to double:  36 years in this case.


Simplify inside the parentheses and then take the natural log of both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(1.001\overline{6}\right)^{12t}\ =\ \ln(2)]


Use the rules of logs:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n)\ =\ n\log_b(x)]


to write:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 12t\ln\left(1.001\overline{6}\right)\ =\ \ln(2)]


Then solve for *[tex \Large t]:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ t\ =\ \frac{\ln(2)}{12\ln\left(1.001\overline{6}\right)}]


The rest is just poking fun at your calculator.


John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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