(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)