## anonymous 5 years ago You have £1,000 in your savings account. At the end of each month, the bank adds an extra 0.2% of interest to the outstanding balance and you make a withdrawal of £10. How much is in the savings account after this has happened for 36 months, rounded to the nearest pound? (I guess you need to solve it with a linear difference equation, but I cannot figure it out)

1. anonymous

$a ^{n}x _{0} +b (a ^{n}-1)/(a-1)$ I found this formula for it and it gives the good answer but I do not see why. (a^n=1.002^36 , xo=1000 , b=-10)

2. anonymous

yeah i have to write this down. certainly cannot do it on the fly. if at all!

3. anonymous

wait, why isn't it just $1000(1.002)^{10}-10(1.002)^9$?

4. anonymous

actually my exponents are ridiculous. it is 36 months not ten. so i think maybe $1000(1.002)^{36}-10(1.002)^{35}$

5. anonymous

i just wrote out the first three months and saw what i got.

6. anonymous

I am not sure here at all, I found that formula and it did give me the correct answer, but I can only see the answer unfortunately

7. anonymous

first month P(1.002) second month (P(1.002)-10)(1.002)=P(1.002)^2-10(1.002) third month (P(1.002)^2-10(1.002))-10)(1.002)=P(1.002)^3-10(1.002)^2-10(1.002)

8. anonymous

oh wait maybe you get a geometric series for the ten part.

9. anonymous

yeah that formula does look like a geometric series

10. anonymous

first part is definitely p(1.002)^36

11. anonymous

lets see what the last one is.

12. anonymous

I think I see now why it is so

13. anonymous

guess it depends on whether you take out ten pounds at the end. if so you will get 10+10(1.002)+10(1.002)^2+...+10(1.002)^35

14. anonymous

$x _{n}=a ^{n} x _{0} +(a ^{n-1} +....+a+1)b$

15. anonymous

and the last part is a series

16. anonymous

equal to a^n -1/a-1

17. anonymous

yeah i think so. i bet this simplifies to get essentially what i wrote at the beginning. if my algebra was good enuf i could do it

18. anonymous

did you try 1000(1.002)^36-10(1.002)^35?

19. anonymous

no, but I can

20. anonymous

that gives a positive amount, that cannot be right, the interest is around 2 and you take out 10

21. anonymous

we can use 1+r+r^2+...+r^35=$\frac{r^{36}-1}{1-r}$ with r = 1.002

22. anonymous

3.7289 rounded

23. anonymous

yes that is what I thought too, but isnt the denominator r-1 ?

24. anonymous

multiply by ten to get 37.289

25. anonymous

yea typo

26. anonymous

denominator in this case is .002

27. anonymous

oh wait my calculation is wrong

28. anonymous

i divided by .02

29. anonymous

the answer was something around 700

30. anonymous

i am off by a decimal

31. anonymous

:) happens

32. anonymous

$\frac{(1.002)^{36}-1}{.002}=37.289$

33. anonymous

according to my calculator.

34. anonymous

multiply by ten to get 372.89

35. anonymous

according to my brain

36. anonymous

subtract from 1000(1.002)^36 = 1074.578 rounded

37. anonymous

38. anonymous

and i get as a final answer (as they say) 701.68

39. anonymous

whew.

40. anonymous

glad we got this worked out even if a couple of false starts

41. anonymous

maths is like this, try hard and fail a lot