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Let's suppose I have $3$ flowers in a field initially and that the number of flowers doubles every month. I can then write that $$N=3(1+0.5/12)^{12t}$$

where $t$ is the time in years. Right? But then if I want to know the number of flowers after $3$ months, which should be the $24$, I use this formula and it does not give me $24$. Why?

Michael Hardy
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p-bromonitrobenzene
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    Actually the number of flowers after $t$ years is $$N(t)=3\cdot2^{12t}.$$ – Did Mar 13 '16 at 22:18
  • Well it's just the parenthesis that doesn't make sense. It should (pardon the expression) clearly be $N = 3 \cdot 2^{12 t}$ as over a year 12 doubling must occur and since there is only one exponential function which does this this must be the one. – Squid Mar 13 '16 at 22:18
  • No I know that this should be the equation but why doesn"t it work using what I used? – p-bromonitrobenzene Mar 13 '16 at 22:21
  • The formula for compound interest that you are using assumes that the growth rate is given as percentage per year, not per month. – Justin Benfield Mar 13 '16 at 22:21
  • Ok so if I want to use this formula to que the correct form I would do $$N=3(1+0.5)^t$$ where t is now in month is that right? – p-bromonitrobenzene Mar 13 '16 at 22:22
  • Because it still doesn't work – p-bromonitrobenzene Mar 13 '16 at 22:23
  • You would need to convert the growth rate of 100% per month to the equivalent growth rate per year. – Justin Benfield Mar 13 '16 at 22:25
  • To do that conversion, note that the compounding (of 100%) of flowers happen monthly, we want it to be measured in yearly terms, and there are 12 months in a year. Hence we have 12 compoundings (of 100% each time) per year, thus the yearly rate, is $r=12$ – Justin Benfield Mar 13 '16 at 22:40

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If $t$ is the number of years, then $12t$ is the number of months, and that is the number of times that one multiplies by $2$. Thus

$$ N= 3\cdot \underbrace{2\cdot2\cdot2\cdots2}_{\begin{smallmatrix} \text{just as many 2s} \\ \text{as the number} \\ \text{of months} \end{smallmatrix}} = 3\cdot 2^{\text{number of months}} = 3\cdot 2^{12t}. $$

After three months, you have $t = \frac 1 4$ (since $3 \text{ months} = \frac 1 4 \text{ year}$).

Michael Hardy
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