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This sounds odd but, there was a math question in my textbook which intrigued me. So it goes ‘Sarah puts $200 in her account every month, which is subject to 1% interest. Show the value of money in her account after x months’. So I assumed that you’d need to use the sigma notation in order to reduce using … . This might be a really odd question but does anyone know the answer of how to represent this formula?

$$x_{n+1}=1.01(x_n+200)$$ $$x_1=0$$ $$x_n$$ represents how much was in the account on the nth month prior to placing the $200 in the account on that month.

x represents the value of money in the bank. Essentially, I’m asking whether there is a simplified method of finding the value of money after y months

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    $\begingroup$ Do some investigating. What did she start with. What did she have after month 1, month 2, and month three. Can you see a pattern? $\endgroup$
    – Steven Alexis Gregory
    Commented Jun 16, 2018 at 16:49
  • $\begingroup$ This has nothing to do with measure theory. $\endgroup$
    – Jakobian
    Commented Jun 16, 2018 at 17:24
  • $\begingroup$ If you edit the question to show us your answer using "..." we may be able to tell you a fancier way to write it. $\endgroup$
    – Ethan Bolker
    Commented Jun 16, 2018 at 17:34

2 Answers 2

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In Month $x=1$, The Account value $A_1$ is $200$.

In Month $x=2$, she puts in an extra $200$, and gets $1\%$ interest on what she already had, so $A_2=402$.

Then it follows $A_{n+1}=1.01(A_n)+200$

What we have here is a recurrence relation. We need to find a closed form from this. $$A_1=200$$

$$A_2=1.01(200)+200$$ $$A_3=1.01(1.01(200)+200)+200)=1.01^2(200)+1.01(200)+200$$ $$A_4=1.01(1.01^2(200)+1.01(200)+(200))+200$$ $$A_4=1.01^3(200)+1.01^2(200)+1.01(200)+200$$

Note that $A_1=200$ and the pattern is easily spotted: $$A_n=200\sum_{k=0}^{n-1}{[(1.01)^k]}$$

Sum notation is indeed required as suggested in your question.

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  • $\begingroup$ Thank you for the answer. Do you know of anywhere that I could learn more about recurrence relations, as I would like to understand what you did to a greater extent? $\endgroup$
    – user491194
    Commented Jun 16, 2018 at 18:22
  • $\begingroup$ No problem. All I did to find the recurrence relation was consider what happened at each step, the previous month's balance was multiplied by $1.01$ to mark the interest, and then $200$ was added. You just need to be able to convert the worded question into a mathematical expression you can use. Does the second part of the answer (From "$A_2=1.01(A_1)+200$" down) make sense? $\endgroup$
    – Rhys Hughes
    Commented Jun 16, 2018 at 18:28
  • $\begingroup$ I understand it up to finding the pattern. I do see that it works, but I’m just not sure how you managed to put it into a sum notation. $\endgroup$
    – user491194
    Commented Jun 16, 2018 at 18:33
  • $\begingroup$ I've changed $A_1$ for $200$ throughout to make it a bit clearer. Do you see how the power of $1.01$ in front of each $200$ term increases by $1$ each time, and then the term $200$ is added on the end? I can use this and that $200=1.01^0(200)$ to form the series I gave. $\endgroup$
    – Rhys Hughes
    Commented Jun 16, 2018 at 18:38
  • $\begingroup$ So based off my sequence for example: $$A_7=200[1.01^6+1.01^5+1.01^4+1.01^3+1.01^2+1.01^1+1.01^0]$$ $\endgroup$
    – Rhys Hughes
    Commented Jun 16, 2018 at 18:39
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The "sigma notation" is just a summation, so you don't use it directly to convey the notion of interest. This question requires you to think a bit more about how interest and deposit timing affects what's in the bank account.

One way to think about it is to imagine that instead of putting each month's deposit into the same account, she puts it into a different account each month, with each account earning the same $1$ percent interest. How much money does she have in the $x$ different accounts after $x$ months? Is it different from what it would be if she put them into the same account?

The summation simply permits you to denote the addition of what's in all the $x$ accounts together.

P.S. On a side note, I have to say that I find the use of $x$ to denote an integer number of months a little irritating. Downvoting the textbook.

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