Annuity Value 0F5B78

1. **State the problem:** We want to find the future value of an annuity where $2000$ is deposited at the end of each year for $20$ years at an interest rate of $5\%$ compounded annually. 2. **Formula for the future value of an ordinary annuity:** $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where: - $P$ is the periodic deposit ($2000$), - $r$ is the interest rate per period ($0.05$), - $n$ is the number of periods ($20$). 3. **Calculate the future value:** $$FV = 2000 \times \frac{(1 + 0.05)^{20} - 1}{0.05}$$ 4. Calculate $(1 + 0.05)^{20}$: $$1.05^{20} = 2.653297705$$ 5. Substitute back: $$FV = 2000 \times \frac{2.653297705 - 1}{0.05} = 2000 \times \frac{1.653297705}{0.05}$$ 6. Simplify the fraction: $$\frac{1.653297705}{0.05} = 33.0659541$$ 7. Multiply by $2000$: $$FV = 2000 \times 33.0659541 = 66131.9082$$ 8. **Find the total amount deposited:** $$\text{Total deposits} = 2000 \times 20 = 40000$$ 9. **Calculate the interest earned:** $$\text{Interest} = FV - \text{Total deposits} = 66131.9082 - 40000 = 26131.9082$$ **Final answers:** - Future value of the annuity: $$66131.91$$ - Interest earned: $$26131.91$$