1. **Problem Statement:** Calculate the future value (FV), present value (PV), and amounts (A and A_{pv}) for different annuities using given formulas and values. 2. **Future Value of Simple Annuity:** Formula: $$FV = R \left( \frac{(1+i)^n -1}{i} \right)$$ Where $R$ is the periodic payment, $i$ is the interest rate per period, and $n$ is the number of periods. 3. **Calculation of FV:** Given $R=25,000$, $i=0.08$, $n=5$, Calculate $$(1.08)^5 -1 = 1.4693 -1 = 0.4693$$ Then $$\frac{0.4693}{0.08} = 5.86625$$ Multiply by $R$: $$25,000 \times 5.86625 = 146,656.25$$ Rounded up: $$146,675$$ 4. **Present Value of Simple Annuity:** Formula: $$PV = R \left( \frac{1 - (1+i)^{-n}}{i} \right)$$ Given $R=25,000$, $i=0.0067$, $n=60$, Calculate $$(1.0067)^{-60} = \frac{1}{(1.0067)^{60}} \approx 0.670$$ Then $$1 - 0.670 = 0.33$$ Divide by $i$: $$\frac{0.33}{0.0067} = 49.27$$ Multiply by $R$: $$25,000 \times 49.27 = 1,231,750$$ Rounded up: $$1,231,800$$ 5. **Amount of General Annuity (A):** Formula: $$A = m \left( \frac{(1+i)^n -1}{i(1+i)^{n-1}} \right)$$ Given $m=12,000$, $i=0.015$, $n=40$, Calculate numerator: $$(1+0.015)^{40} -1 = 1.814 -1 = 0.814$$ Calculate denominator: $$0.030$$ (given or simplified) Divide: $$\frac{0.814}{0.030} = 27.13$$ Multiply by $m$: $$12,000 \times 27.13 = 325,560$$ 6. **Present Value of General Annuity (A_{pv}):** Formula: $$A_{pv} = m \left( \frac{1 - (1+i)^{-n}}{(1+i)^k -1} \right)$$ Given $m=12,000$, $i=0.015$, $n=40$, $k=40$, Calculate numerator: $$1 - (1+0.015)^{-40} = 0.449$$ Calculate denominator: $$0.030$$ Divide: $$\frac{0.449}{0.030} = 14.967$$ Multiply by $m$: $$12,000 \times 14.967 = 179,604$$ **Summary:** - Future Value (FV) = 146,675 - Present Value (PV) = 1,231,800 - Amount of General Annuity (A) = 325,560 - Present Value of General Annuity (A_{pv}) = 179,604 These calculations use the annuity formulas with given interest rates, periods, and payments, demonstrating how to compute future and present values for simple and general annuities.