You are expecting $22$ growing annual payments, starting with $$1,000$ next year. The payments will increase by $5\%$ from year to year. The appropriate discount rate is $5\mathrm{.}6\%.$

a$.$ Calculate the present value of those expected payments, using the annuity formula.

b$.$ Calculate the present value of those expected payments, by forecasting each of the cash flows, computing their present values separately, and then computing the sum of the present values.