Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r. For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest. Denote by An the amount of money in your account in the nth month. 1. Explain why A₁ = m dollars. 2. Explain why A₂ = m+m(1 + r) dollars. 3. Write down explicit expressions for A3 and A4. This is the crucial step. 4. Explain why An = m+m(1 + r) + m(1+r)²+...+m(1+r)n-¹ dollars. 5. Use the formula for a geometric sum to show that An = m (1+r)" - 1 T dollars. 6. If your account has a monthly interest rate r = 0.002 and you deposit $200 monthly for 5 years, how much money will you have in your account after the 5 years? (Hint: How many months?)

Compute and compare the following interests situations if you deposit $2500 into an account with 3.5% interest for 10 years. 20) The account pays simple interest. 21) The account pays compounding interest monthly. 22) The account pasy continuously compounding interest. 23) How much difference is there between the three different types of interest? Explain.

For each of the following situations involving annuities, solve for the unknown. Assume that interest is compounded annually and that all annuity amounts are received at the end of each period. (i = interest rate, and n = number of years) (FV of $1, PV of $1, FVA of $1, PVA of $1, FVAD of $1 and PVAD of $1) (Use appropriate factor(s) from the tables provided. Round your final answers to nearest whole dollar amount.)       Present Value Annuity Amount i = n = 1. ? $2,400 8% 5 2. 533,082 140,000 ? 4 3. 583,150 180,000 9% ? 4. 530,000 75,502 ? 8 5. 235,000 ? 10% 4

For each of the following situations involving annuities, solve for the unknown. Assume that interest is compounded annually and that all annuity amounts are received at the end of each period. (i=interest rate, and n=number of years)(FV of $1, PV of $1, FVA of $1, PVA of $1, FVAD of 1$ and PVAD of $1) (Use appropriate factor (s) from the tables provided. Round your final answers to nearest whole dollar amount.)                   Present Value            Annuity Amount         i=                 n= ______________          $                 2,600           8%                5 507,866                                        135,000        _____              4 661,241                                        170,000            9%             ____ 540,000                                           78,557        _____              8 230,000                          _____________           10%                4