## anonymous one year ago 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.

1. anonymous

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

2. anonymous

1123.6 is what im guessing, dont quote me lol

3. anonymous

what?

4. anonymous

don't give out an answer. 0_o

5. anonymous

and im pretty sure its wrong

6. anonymous

so the annuity is 20$and the intrest is compounded monthly for 4 years 7. anonymous the first thing you would do is get the intrest plus the principle 8. anonymous and then add 20 for 4 years 9. anonymous you could use this website here to find out the interest + principal http://www.calculatorsoup.com/calculators/financial/ 10. anonymous Im not interested in calculators, I want to make that into a recursive equation and then solve the recurisve equation 11. anonymous Y_{0}=100$ 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....

12. anonymous

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.

13. anonymous

@IrishBoy123 do you anything about recursive equations?

14. IrishBoy123

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

15. IrishBoy123

Example 2.2 looks a bit like yours

16. anonymous

$Y _{n+1}-1,06Y _{n}-200=0$ 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$

17. anonymous

$C _{1}=24.53$

18. anonymous

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

19. anonymous

and I ment that A must be equal to 333.3

20. anonymous

@IrishBoy123 check this out now

21. IrishBoy123

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

22. IrishBoy123

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

23. IrishBoy123

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

24. anonymous

oops forgot the exponent over r in the formula!

25. anonymous

he should have 6786.36 dollars after 4 years?

26. anonymous

@IrishBoy123

27. IrishBoy123

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