Answer:
Francesco can achieve this amount by investing at an APR of 7.14% compounded annually , or 7% compounded every six months or 6.95% compounded quarterly
Step-by-step explanation:
This question can be solved using the concept of compounded interest
The formula for amount obtained when a principal of p was invested at a rate r compounding n times a year is:
[tex]Amt = p(1+\frac{r}{100n})^n[/tex]
Here p = 70000, amt = 75000,
So when compounded annually, n=1
[tex]Amt =p(1+\frac{r}{100n})^n\\75000=70000(1+\frac{r}{100})^1\\\frac{15}{14}=1+\frac{r}{100}\\\frac{1}{14}=\frac{r}{100}\\r = \frac{100}{14}=7.14[/tex]
For compounding every 6 months n=2
[tex]Amt =p(1+\frac{r}{100n})^n75000=70000(1+\frac{r}{200})^2\\\frac{15}{14}=(1+\frac{r}{200})^2\\\sqrt{\frac{15}{14}}=1+\frac{r}{200}\\r = 200*(\sqrt{\frac{15}{14}}-1)=7.01[/tex]
For compounding quarterly n=4
[tex]Amt =p(1+\frac{r}{100n})^n\\75000=70000(1+\frac{r}{400})^4\\\frac{15}{14}=(1+\frac{r}{400})^4\\(\frac{15}{14})^\frac{1}{4}=1+\frac{r}{400}\\r = 6.95[/tex]
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