1. **Problem statement:** You deposit 1000 at the start of every quarter into an account with an annual interest rate of 2.5%, compounded monthly. We want to find the total amount in the account after 4 years and 9 months. 2. **Key information:** - Quarterly deposit: 1000 - Annual interest rate: 2.5% = 0.025 - Compounded monthly means interest is added 12 times a year. - Total time: 4 years 9 months = 4.75 years 3. **Formula for future value of an annuity with compound interest:** $$FV = P \times \frac{(1 + r/n)^{nt} - 1}{(1 + r/n)^{m} - 1}$$ where: - $P$ = deposit amount per period - $r$ = annual interest rate - $n$ = number of compounding periods per year - $t$ = total years - $m$ = number of compounding periods per deposit period 4. **Calculate parameters:** - $P = 1000$ - $r = 0.025$ - $n = 12$ (monthly compounding) - $t = 4.75$ - Deposits are quarterly, so each deposit period is 3 months = 3 compounding periods, so $m = 3$ 5. **Calculate $(1 + r/n)$:** $$1 + \frac{0.025}{12} = 1 + 0.0020833333 = 1.0020833333$$ 6. **Calculate total compounding periods:** $$nt = 12 \times 4.75 = 57$$ 7. **Calculate numerator:** $$ (1.0020833333)^{57} - 1 $$ Calculate power: $$ (1.0020833333)^{57} \approx 1.1247 $$ So numerator: $$1.1247 - 1 = 0.1247$$ 8. **Calculate denominator:** $$ (1.0020833333)^3 - 1 $$ Calculate power: $$ (1.0020833333)^3 \approx 1.00626 $$ So denominator: $$1.00626 - 1 = 0.00626$$ 9. **Calculate fraction:** $$\frac{0.1247}{0.00626} \approx 19.92$$ 10. **Calculate future value:** $$FV = 1000 \times 19.92 = 19920$$ **Answer:** After 4 years and 9 months, the account balance will be approximately **19920**.