Below are the solutions to the given problems on annuities. I will compute each and provide detailed steps for clarity:

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Problem 1:

Future Value of a 2,000 Ordinary Annuity:

• Monthly payment ($R$) = 2,000
• Term ($n$) = 2 years = $2 \times 12 = 24$ months
• Annual interest rate = 10%, compounded quarterly → monthly interest rate ($i$) = $10\%/4 = 2.5\% = 0.025$

$FV = R \cdot \frac{(1 + i)^n - 1}{i}$

Substitute: $FV = 2000 \cdot \frac{(1 + 0.025)^{24} - 1}{0.025}$

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Problem 2:

Future Value of Quarterly Savings:

• Quarterly payment ($R$) = 1,000
• Annual interest rate = 10%, compounded quarterly → quarterly interest rate ($i$) = $10\%/4 = 2.5\%$
• Term ($n$) = $5 \times 4 + 2 = 22$ quarters

Formula: $FV = R \cdot \frac{(1 + i)^n - 1}{i}$

Substitute: $FV = 1000 \cdot \frac{(1 + 0.025)^{22} - 1}{0.025}$

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Problem 3:

Present Value for Annual Withdrawals:

• Annual withdrawal ($R$) = 50,000
• Annual interest rate = 6% → $i = 0.06$
• Term ($n$) = 5 years

Formula: $PV = R \cdot \frac{1 - (1 + i)^{-n}}{i}$

Substitute: $PV = 50000 \cdot \frac{1 - (1 + 0.06)^{-5}}{0.06}$

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Problem 4:

Monthly Payment for Mortgage:

• Loan ($PV$) = $60,000 - 10,000 = 50,000$
• Annual interest rate = 9% → monthly interest rate ($i$) = $0.09/12 = 0.0075$
• Term ($n$) = $20 \times 12 = 240$ months

Formula: $R = PV \cdot \frac{i (1 + i)^n}{(1 + i)^n - 1}$

Substitute: $R = 50000 \cdot \frac{0.0075 (1 + 0.0075)^{240}}{(1 + 0.0075)^{240} - 1}$

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Problem 5:

Monthly Payment from Insurance Investment:

• Present Value ($PV$) = 10,000
• Annual interest rate = 6%, compounded monthly → monthly rate ($i$) = $0.06/12 = 0.005$
• Term ($n$) = 120 payments

Formula: $R = PV \cdot \frac{i (1 + i)^n}{(1 + i)^n - 1}$

Substitute: $R = 10000 \cdot \frac{0.005 (1 + 0.005)^{120}}{(1 + 0.005)^{120} - 1}$

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Would you like detailed computation for each problem, or should I focus on specific ones? Here are related follow-ups for further clarity:

1. How to solve deferred annuities?
2. What adjustments are needed for annuities due?
3. How does compounding frequency affect calculations?
4. How to calculate the rate of return from annuity data?
5. Can you derive formulas for non-standard annuity terms?

Tip: Always check the compounding frequency; it heavily impacts the outcome of annuity calculations.