Jimmy invests $\$ 17,000$ in an account that pays $5.18 \%$ compounded quarterly. How long (in years and months) will it take for his investment to reach $\$ 20,000 ?$
years and months
(Round the answer for months up to the nearest multiple of three.)

Step 1 ：Given that Jimmy invests $17,000 in an account that pays 5.18% compounded quarterly, we need to find out how long it will take for his investment to reach $20,000.

Step 2 ：We use the formula for compound interest, which is \(A = P(1 + \frac{r}{n})^{nt}\), where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

Step 3 ：In this case, we know that A = $20,000, P = $17,000, r = 5.18% = 0.0518, and n = 4 (since the interest is compounded quarterly). We need to solve for t.

Step 4 ：We can rearrange the formula to solve for t: \(t = \frac{\log(\frac{A}{P})}{n \cdot \log(1 + \frac{r}{n})}\).

Step 5 ：Substituting the given values into the formula, we get \(t = \frac{\log(\frac{20000}{17000})}{4 \cdot \log(1 + \frac{0.0518}{4})}\).

Step 6 ：Solving this equation gives us \(t = 3.1577023729171936\).

Step 7 ：Since we need the time in years and months, we round the decimal part of the year up to the nearest multiple of three to get the number of months. This gives us 3 years and 3 months.

Step 8 ：Final Answer: It will take approximately \(\boxed{3}\) years and \(\boxed{3}\) months for Jimmy's investment to reach $20,000.