Sigma notation for repetitions of 1.01(n+200)

Question

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

Answer 1

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.

Answer 2

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.

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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.