Suppose $1,500 is compounded weekly for 46 years. If no other deposits are made, what rate is needed for the balance to triple in that time?

Answer:The rate is needed is 1.037%.Step-by-step explanation:Given : Suppose $1,500 is compounded weekly for 46 years. If no other deposits are made.To find : What rate is needed for the balance to triple in that time?Solution : Applying compound interest formula,[tex]A=P(1+\frac{r}{n})^{nt}[/tex]Where, P is the principalA is the amount The balance to triple in that time i.e. A=3Pr is the rate t is the time t=46 yearsCompounded weekly so n=52Substitute the value in the formula,[tex]3P=P(1+\frac{r}{52})^{52\times 46}[/tex][tex]3=(1+\frac{r}{52})^{2392}[/tex]Taking log both side,[tex]\log 3=2392\ log(1+\frac{r}{52})[/tex][tex]\frac{\log 3}{2392}=\ log(1+\frac{r}{52})[/tex][tex]0.00019946=\ log(1+\frac{r}{52})[/tex]Taking exponential both side,[tex]e^{0.00019946}=1+\frac{r}{52}[/tex][tex]1.000199-1=\frac{r}{52}[/tex][tex]0.000199=\frac{r}{52}[/tex][tex]r=0.000199\times 52[/tex][tex]r=0.010372[/tex]Into percentage,[tex]r=0.010372\times 100[/tex][tex]r=1.0372[/tex]Therefore, the rate is needed is 1.037%.