Problem
Guy invests $S c$ at the beginning of each quarter into an account earming interest at a rate of $4 \%$ per annum, compounding quarterly, where $c$ is your assigned value in the table above. The amount in the account immediately after the $n^{\text {th }}$ quarter can be determined using the recurrence relation \[ A_{n}=A_{n-1}(1.01)+c(1.01) \] where $n=1,2,3, \ldots$ and $A_{0}=0$ (a) Use the recurrence relation to find the amount of money in the account at the end of 2 the third quarter. (b) Calculate the amount of interest earned in the first three quarters. 1 (c) Calculate the amount of money in the account at the end of 10 years. 2
Solution
Step 1 ：A_1 = A_0(1.01) + c(1.01) = 0(1.01) + c(1.01) = c(1.01)
Step 2 ：A_2 = A_1(1.01) + c(1.01) = c(1.01)(1.01) + c(1.01) = c(1.01)^2 + c(1.01)
Step 3 ：A_3 = A_2(1.01) + c(1.01) = (c(1.01)^2 + c(1.01))(1.01) + c(1.01) = c(1.01)^3 + c(1.01)^2 + c(1.01)
Step 4 ：\boxed{A_3 = c(1.01)^3 + c(1.01)^2 + c(1.01)}
Step 5 ：I_3 = A_3 - 3c = c(1.01)^3 + c(1.01)^2 + c(1.01) - 3c
Step 6 ：\boxed{I_3 = c(1.01)^3 + c(1.01)^2 + c(1.01) - 3c}
Step 7 ：A_{40} = c(1.01)^{40} + c(1.01)^{39} + \cdots + c(1.01)^2 + c(1.01)
Step 8 ：\boxed{A_{40} = c(1.01)(1.01^{40} - 1)/(1.01 - 1)}
