Question

The first day of a new year Adam opens a bank account and deposit 100 dollars . Then
he deposits 20 dollars every moth with the interest rate, 6 % that is calculated monthly.
How much money does Adam have on his account exactly 4 years after he opened the account ? Set up a proper recursive equation and solve this.

Answer 1

Need help with the recursive equation and how to think, im all out of ideas here.

Answer 2

1123.6 is what im guessing, dont quote me lol

Answer 3

what?

Answer 4

don't give out an answer. 0_o

Answer 5

and im pretty sure its wrong

Answer 6

so the annuity is 20$ and the intrest is compounded monthly for 4 years

Answer 7

the first thing you would do is get the intrest plus the principle

Answer 8

and then add 20 for 4 years

Answer 9

you could use this website here to find out the interest + principal http://www.calculatorsoup.com/calculators/financial/

Answer 10

Im not interested in calculators, I want to make that into a recursive equation and then solve the recurisve equation

Answer 11

Y_{0}=100$
Y_{1}=100*1.06+20=126$
Y_{2}=126*1.06+20=153.56$

Y_{0}=100$
Y_{1}=Y_{0}*1.06+200
Y_{2}=Y_{1}*1.06+200

that is the recursive sequence

so then you have
\[Y _{n+1} = 1.06Y _{n}+200\]

and then solve that one....

Answer 12

well i don't know much there because my finance teacher always let us use calculators but there are the equations there in calculator soup. That's all i can help you with then sorry.

Answer 13

@IrishBoy123 do you anything about recursive equations?

Answer 14

what you've built seems right

http://www.math.kth.se/math/GRU/2012.2013/SF1610/CINTE/mastertheorem.pdf
re-solving, some techniques here, but nothing that I am au fait with, would need to read it myself :-)

happy to try help if you are in need

Answer 15

Example 2.2 looks a bit like yours

Answer 16

\[Y _{n+1}-1,06Y _{n}-200=0\] is our homogenous equation

and to solve that you make a characteristic equation:
\[r-1.06=0\]
which means that our root must be r=1.06
and then we guess a particular equation of the form A
and if we put that in the equation it will be

\[A-1.06A=0\]

\[A(1-1.06)=0\]
\[0.06A=200\]
\[A=\frac{ 200 }{ 0.06 } = 3333.3\]
So A must be equal to 3333.3

then we have to make the common solution
\[y _{n}=(C _{1}*n+C _{2})*1.06-3333\]
as we know
\[y _{0}=100\]
and
\[y _{1}=126\]

so try that with the common solution
\[y _{0}=(C_{1}*0 + C _{2})*1.06 - 333.3 = 100\]
\[C _{2}=408.77\]
\[y _{1}=(C _{1}*1+C _{2})*1.06-333.3=126\]
\[C _{1}=\frac{ -1.06C _{2}+459.3 }{ 1.06}\]
\[C _{1}=\frac{ -1.06*408.77+459.3 }{ 1.06}\
\[C _{1}=24.53

the solution we were looking for is
\[Y _{n}=(24.53*n+408.77)*1.06-333\]

Answer 17

\[C _{1}=24.53\]

Answer 18

and the solution we are looking for is
\[Y _{n}=(24.52*n+408.77)*1.06-333\]

Answer 19

and I ment that A must be equal to 333.3

Answer 20

@IrishBoy123 check this out now

Answer 21

seems *slightly* out. and i would expect to see a \(1.06^n\) in there for sure to get the compounding

have put them in a spreasdheet - attached for checking pruposes

Answer 22

ouch, might be annual 6% so 6/12 monthly.....

Answer 23

this is what i mean by exponents....
https://gyazo.com/8e485926c771572c6f243e742837f94d

Answer 24

oops forgot the exponent over r in the formula!

Answer 25

he should have 6786.36 dollars after 4 years?

Answer 26

@IrishBoy123

Answer 27

no, my bad

look at the second pdf of same name.

i think the 6% is annual so you apply 1/12 [not strictly true] of that per month

strictly speaking you break 6% pa into monthly by saying
\[ (1+r_m)^{12} = 1 +0.06\]
\[ (1+0.06)^{\frac{1}{12}} - 1 = r_m\]
gives 0.4867551% monthly interest, whereas

\[\frac{6\%}{12} = 0.5\%\]

that will make a difference to how you do this