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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