Question

6) Suppose you save money at a bank that pays 3% per year in simple interest. If you open the account with $1000, and in addition to the interest that the bank pays, on the last day of each year, you deposit $200 more into the account. Give a recurrence relation for S(n) your money at the end of n years, forn 20. How much money will you have at the end of 4 years?

Answer 1

Here, S(1) = 1000*103%

i.e., S(1) = 1000*1.03

And, S(2) = [S(1)+200]*1.03

And, S(3) = [S(2)+200]*1.03

Therefore, the recurrence relation is : $\boldsymbol{S(n+1)=[S(n)+200]*1.03}$

Putting n = 3 we get,

$S(3+1)=[S(3)+200]*1.03$

i.e.,

S(4) = [S(3) +200) * 1.03

i.e.,

S(4) = [[S(2) +200) * 1.03 + 200] * 1.03

i.e., $S(4)=[[[S(1)+200]*1.03+200]*1.03+200]*1.03$

i.e.,

S(4) = [[[1000 * 1.03 + 200 * 1.03 + 200] * 1.03 + 200 * 1.03

i.e., $S(4)=[[[1030+200]*1.03+200]*1.03+200]*1.03$

i.e., $S(4)=[[1230*1.03+200]*1.03+200]*1.03$

i.e., $S(4)=[[1266.9+200]*1.03+200]*1.03$

i.e., $S(4)=[1466.9*1.03+200]*1.03$

i.e., $S(4)=[1510.907+200]*1.03$

i.e., $S(4)=1710.907*1.03$

i.e., $S(4)=1762.23421$

i.e., $S(4)\approx1762.23$

Therefore, at the end of 4 years, I will have $1762.23.