a person invests 1000 at a bank at 4 percent compound interest compounded annually and every year government and bank charges amounting to C are deducted and if An is the value of the investment at the end of 10 years.
Solve this difference equation.
if i)C=0 ii)C=40
$ A_{10} = 1.04 A_{n-1} - C $
$ A_0 = 1000 $
i) C=0
$ A_{10} = 1.04^{10} . 1000 = 1480.24 $
ii) C=40
$ A_n = 1.04 . A_{n-1} - 40 $
Particular solution: Put $ A_n = A_{n-1} = A* $
$ A* = 1.04 . A* - 40 $
$ A* = 40/0.04 = 1000 $
General solution of the associated homogenous equation: $ A_n = 1.04 A_{n-1} $
$ a_n = A . 1.04^n $
General solution of the difference equation: $ A_n = A_n + A* $
$ A_n = A . 1.04^n + 1000 $
$ A_0 = A . 1.04^0 + 1000 $
$ A = 1000 $
$ A_{10} = 1000 . 1.04^{10} + 1000 $
$ = 1480.20 + 1000 $
$ = 2480.24 $
Getting 2480.24 as the answers tell me something is wrong. But I cant figure out which part of my working is incorrect.
Since 1000-40 for 10 years I believe the answer should be
$ A_{10} = 1.04^{10} . 1000 - 40 $
Please advise.