a Calculate a_2, a_3, and find a recursive pattern.
B
b Use the recursive equation from Part A and a spreadsheet.
A
aRecursive Equation: a_n=1.08(a_(n-1)-30 000)
B
b a_(n-1)=a_n/1.08+30000, a_0=318 107.98. See solution for the explanation.
Practice makes perfect
a After (n-1) years your balance is a_(n-1). At the beginning of the n^(th) year you withdraw $30 000. This tells us that the balance after withdrawing is a_(n-1)-30 000. During one year you can expect to earn an annual return of 8 %=0.08 on your savings. Therefore, the balance after n years is 1.08(a_(n-1)-30 000).
a_n=1.08(a_(n-1)-30 000)
b First, we are asked to solve the recursive equation from Part A for a_(n-1). Let's do it!
Next, we are asked to find a_0, the minimum amount of money you should have in your account when you retire. This tells us that after 20 years in retirement the balance should be 0. Therefore, a_(20)=0. Now, we will use the recursive equation from Part A and a spreadsheet to find a_0.
a_(20)=0, a_n=1.08(a_(n-1)-30 000)
Let's enter Year in cell A1, Balance in cell B1, 20 in cell A2, and 0 in cell B2.
Write =A2-1 in cell A3, and write =Round(B2/1.08+30000,2) in cell B3. When we hit enter, we will get the following.
Next, highlight cells A3 and B3, copy them, highlight cells A4 through B22, and paste.
From the spreadsheet we can notice that a_0=318 107.98. This tells us that the minimum amount of money you should have is $318 107.98.
