SOLUTION: How long does it take for an investment to double in value if it is invested at 4​% compounded quarterly question mark  Compounded​ continuously?

     (1)

where

F = future value

P = present value

r = annual interest rate

n = number of payments per year

t = number of years





This problem asks to find t for F=2P,  r=4% (0.04), and n=4 interest payments per year:

 

Cancel P from both sides, and simplify where possible:



Take ln() of both sides:



Solve for t:

 years



 years  (about 17yrs, 151.6days}





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For continuous compounding:

   (2)



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An aside: formula (2) can be derived by taking the limit of (1) as n goes to infinity:

     

   Replace r/n by 1/(n/r) and raise the inner expression first to the 'n/r'

     power, and then to the 'rt' power (so the r's would cancel if we carried

     it out):

  

   The green expression has a limit of 'e' as n-->infinity:

  

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 years (about 17yrs, 120days)