1. **Problem Statement:** Gary deposits 3129.34 at the beginning of every month for 4 years to buy a motorcycle. The fund is compounded quarterly at 5%. We need to find the total amount he will have after 4 years. 2. **Formula Used:** Since deposits are monthly but compounding is quarterly, we use the formula for the future value of an annuity due with periodic compounding: $$FV = P \times \frac{(1+i)^n - 1}{i} \times (1+i)$$ where: - $P$ is the monthly deposit, - $i$ is the effective interest rate per deposit period, - $n$ is the total number of deposits. 3. **Calculate the effective interest rate per month ($i_2$):** Given annual nominal rate compounded quarterly $r=0.05$, quarterly rate $i_q=\frac{0.05}{4}=0.0125$. Effective monthly rate $i_2$ satisfies: $$ (1+i_2)^3 = 1 + i_q $$ $$ i_2 = (1+0.0125)^{\frac{1}{3}} - 1 $$ Calculate: $$ i_2 = 1.0125^{0.333333} - 1 = 0.004150 $$ (rounded to 6 decimals) 4. **Calculate total number of deposits:** $$ n = 4 \text{ years} \times 12 \text{ months/year} = 48 $$ 5. **Calculate future value:** $$ FV = 3129.34 \times \frac{(1+0.004150)^{48} - 1}{0.004150} \times (1+0.004150) $$ Calculate: $$ (1+0.004150)^{48} = 1.219391 $$ $$ \frac{1.219391 - 1}{0.004150} = 52.521927 $$ $$ FV = 3129.34 \times 52.521927 \times 1.004150 = 165,045.68 $$ 6. **Final Answer:** Rounded to nearest hundredths: 165045.68