a Let a_n be the amount of money in a deposit at the end of the n^(th) year. Mr. Edwards and his company deposit $20 000 into his retirement account at the end of each year. Therefore, a_1=20 000. The account earns 8 %=0.08 interest before each deposit.
a_n- money at the end of then^(th)year
Let's analyze how much money there will be after n years, a_n, for n> 1, in terms of a_(n-1). The balance at the end of the (n-1)^(th) year is a_(n-1). First we add interest, 0.08a_(n-1). Then, we add 20 000 dollars. Therefore, we get the following equation.
a_n= a_(n-1)+ 0.08a_(n-1)+20 000 ⇓
a_n=1.08a_(n-1)+20 000
Finally, we can write a recursive formula for a_n.
Recursive Rule:
a_1=20 000,
a_n=1.08a_(n-1)+20 000forn>1
b We are asked to calculate the first eight terms of a_n. We will use the recursive formula from Part A.
n
a_n=1.08a_(n-1)+20 000
1.08a_(n-1)+20 000
a_n
1
a_1=20 000
--
a_1= 20 000
2
1.08a_1+20 000
1.08( 20 000)+20 000
a_2= 41 600
3
1.08a_2+20 000
1.08( 41 600)+20 000
a_3= 64 928
4
1.08a_3+20 000
1.08( 64 928)+20 000
a_4=90 122.24
5
1.08a_4+20 000
1.08(90 122.24)+20 000
a_5=117 332.02
6
1.08a_5+20 000
1.08(117 332.02)+20 000
a_6= 146 718.58
7
1.08a_6+20 000
1.08( 146 718.58)+20 000
a_7= 178 456.07
8
1.08a_7+20 000
1.08( 178 456.07)+20 000
a_8=212 732.55
Extra
Calculations Using a Spreadsheet
Let's enter a_1=20 000 in cell A1 and write =Round((1.08)*(A1)+20000,2) in cell A2. When we hit enter, we will get the following.
Next, copy cell A2, highlight cells A3 through A8, and paste.
From the spreadsheet we can write the first eight terms of the given sequence.
