Determine the time necessary for P dollars to double when it is invested at interest rate r compounded annually, monthly, daily, and continuously.

Question

Answer 1

P/P₀ = (1+r)^t with t in years for yearly compounding

Generalized equation for compounding in n pieces of the year:

P/P₀ =(1+r/n)^nt

Now solving generally for the doubling time in years:

2 = (1+r/n)^nt

ln2 = ntln(1+r/n)

t = ln2/(nln(1+r/n))

For continuous:

P/P₀ = e^rt so

ln2 = -rt

t = ln2/r

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