Consider the following two saving schemes. Option 1: On 31 December of year 0, a grandmother opens a bank account with 1,700$ for her daughter. On 31 December of each year (excluding year 0), she pays ’interest’, that is,she deposits an amount that corresponds to 2.5% of the account balance on that day into the account. Further, in each year (excluding year 0), immediately after the ’interest payment’ on 31 December, she deposits an additional 130$ to the account. Let An denote the amount of money in the account on 31 December of year n (after interest payment and deposit). Option 2: On 31 December of year 0, a grandmother opens a bank account with 1,700$ for her granddaughter. Starting in year 1, she deposits 125$ into this account on 1 January of every year. On 31 December of each year (excluding year 0), she pays ’interest’, that is,she deposits an amount that corresponds to 2.5% of the account balance on that day to the account. Let Bn denote the amount of money in the account on 31 December of year n (at close of business, that is, after the interest was paid). (a) Find a recursive definition for An. (draw a timeline.) (b) Find a recursive definition for Bn. (draw a timeline.)
Unitary Method
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Consider the following two saving schemes.
Option 1: On 31 December of year 0, a grandmother opens a bank account with 1,700$ for her daughter. On 31 December of each year (excluding year 0), she pays ’interest’, that is,she deposits an amount that corresponds to 2.5% of the account balance on that day into the account. Further, in each year (excluding year 0), immediately after the ’interest payment’ on 31 December, she deposits an additional 130$ to the account. Let An denote the amount of money in the account on 31 December of year n (after interest payment and deposit).
Option 2: On 31 December of year 0, a grandmother opens a bank account with 1,700$ for her granddaughter. Starting in year 1, she deposits 125$ into this account on 1 January of every year. On 31 December of each year (excluding year 0), she pays ’interest’, that is,she deposits an amount that corresponds to 2.5% of the account balance on that day to the account. Let Bn denote the amount of money in the account on 31 December of year n (at close of business, that is, after the interest was paid).
(a) Find a recursive definition for An. (draw a timeline.)
(b) Find a recursive definition for Bn. (draw a timeline.)
(c) Write a MATLAB program that computes An and Bn for n = 0, 1, 2, . . . , 12, and that displays the values in three columns: n, An, Bn with appropriate headings. Include a printout of the output with your solution.
(d) Use MATLAB to create a plot showing An and Bn as a function of n for years 1 to 12. Label axes appropriately. Include a printout of the plot with the solution.
(e) Adapt MATLAB program so that it computes how much total interest is accrued for each of the two schemes in the years n ≤ 12, and what the average amount of interest isthat is paid for each scheme per year over the years n ≤ 12. Provide the average amount of yearly interest for each of the two schemes.
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