r = 7% Find the time necessary for $1000 to double when it is invested at a rate of r compounded (a) anually, (b) monthly, (c) daily, and (d) continuously

Formula for compounding n times per year A=P(1+r/n)^(nt)
Formula for compounding continuously A=Pe^(rt)
A=Final Amount
P=Initial Amount
r=rate of investment expressed as a decimal
n=number of compoundings per year
t= time in years
 
a) r=7%  n=1 (annually)
A=P(1+r/n)^(nt)
2000=1000(1+.07/1)^(1*t)
2=1.07^t
ln(2)=tln(1.07)
ln(2)/ln(1.07)=t
10.24=t
Final answer: 10.24 years
 
b) r=7% n=12 (monthly)
A=P(1+r/n)^(nt)
2000=1000(1+.07/12)^(12*t)
2=1.0058^(12t)
ln(2)=12tln(1.0058)
ln(2)/[12ln(1.0058)]=t
9.93=t
Final Answer: 9.93 years
 
c) r=7%  t=365 (daily)
A=P(1+r/n)^(nt)
2000=1000(1+.07/365)^(365*t)
2=(1.00019)^(365t)
ln(2)=365tln(1.00019)
ln(2)/[365ln(1.00019)]=t
9.90=t
Final answer: 9.90 years
 
d) r=7% compounded continously
A=Pe^(rt)
2000=1000e^(.07*t)
2=e^(.07t)
ln(2)=.07tlne
ln(2)/[.07lne]=t
9.90=t
Final answer: 9.90 years

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