Question

If you invest $1000 with 5% interest annually but you immediately withdraw 100 after the interest is paid.
a) find the recursive formula for investment
b) find the investment after the third year
c) find formula for a_n
d) when the investment =0
Please, help

Answer 1

for a) \(I_0 = 1000\\I_1 = 1.05 I_0 -100\\I_2= 1.05 I_1 -100\\I_3= 1.05 I_2 -100\\I_4 = 1.05 I_3-100\\-------\\I_n = 1.05 I_{n-1}-100\)

Answer 2

for b) I got 894.... but it is easy to find out
I stuck at c and d
for c) I worked back ward and got the weird formula in which I can't calculate d :(

Answer 3

for c, are you getting
\[a(n)=2000-1000*1.05^n\]

?

Answer 4

If so, setting that equal to 0 and solving \(n\) is easy right ?

Answer 5

how? a(n) = what you showed?

Answer 6

your answer for part b should be 842.375

http://www.wolframalpha.com/input/?i=Table%5B2000-1000*1.05%5En%2C%7Bn%2C0%2C5%7D%5D

Answer 7

yes, I need c and d. :)

Answer 8

we want to solve
\[a_n-1.05a_{n-1}=-100\]

Answer 9

Notice that it is a non homogeneous recurrence relation

Answer 10

so the complete solution will be of form : `particular` + `null`

Answer 11

lets find the null solution first

Answer 12

for `null solution`, we solve
\[a_n-1.05a_{n-1}=0\]

Answer 13

for `null solution`, we solve
\[a_n-1.05a_{n-1}=0\tag{1}\]

suppose the solution is of form \(a_n = c*r^n\)
plugging this in \((1)\) gives
\(c*r^n - 1.05c*r^{n-1}=0\)
\(r-1.05 = 0\)
\(r=1.05\)

so the `null solution` is \(c*1.05^n\)

Answer 14

see if that looks okay so far

Answer 15

next we need to find a particular solution

Answer 16

oh, I didn't think that way. Ok, let me try. Thanks a lot

Answer 17

this `particular` + `null` thingy shows up everywhere you see a `linear` equation

Answer 18

you must have seen this pattern while solving linear diophantine eqations using reverse euclid method

Answer 19

oh, no, it is not my diophantine equation, this is the problem from a student I tutored. I have to solve this problem base on his background, not my background. If I tell something about diophantine, he won't get.

Answer 20

you don't need to tell him anything about diophantine

Answer 21

got you :)

Answer 22

how are you going to find a particular solution ?

Answer 23

I need find and A such that A - 1.05 A =100
hence A = 100/ -0.5

Answer 24

Looks good except for a typo

Answer 25

I need find and A such that A - 1.05 A =-100
hence A = 100/ 0.5

Answer 26

oh

Answer 27

the typo is in first line

Answer 28

yup, -100 , not 100

Answer 29

ok, hence A = 200
and the whole thing is a(n) = C*(1.05)^n +200 and for a(0) =1000, we solve for C, right?

Answer 30

I need find and A such that A - 1.05 A =-100
hence A = 100/ 0.05 = 2000

Answer 31

:) my bad

Answer 32

ok, hence A = 2000
and the whole thing is a(n) = C*(1.05)^n +2000 and for a(0) =1000, we solve for C, right?

Answer 33

Yes, I got what you got. and d is easy now. :)

Answer 34

Thank you so so so much.

Answer 35

np:)