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I'm looking at this recurrence relation problem regarding interest that is compounded monthly. Basically, you deposit $1,000$ into a 5% interest bank account that is compounded monthly. So, after 2 months your account has $1,008.35. It asks you to set up a recurrence relation for the amount of money x(t+1), given that we know the amount of money in the account after t time steps x(t).

Here is my question: What do they mean by the amount of money x(t+1)? What is this supposed to represent in the equation? I thought I would be using x(t-1), which makes more sense to me. I'm not really sure what x(t+1) is supposed to be referring term in terms of the recurrence relation.

Allison
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    They mean a function $f$ that calculates $x(t+1) = f(x(t))$. If $t$ is September then $t+1$ would be October, and you would calculate your next month balance $x(t+1)$ based on the current balance $x(t)$. – dxiv Sep 27 '16 at 07:01
  • This makes sense, but not I'm at a loss for how to set up the recurrence relation. This is what I think: x(t) is my initial condition of $1,000 in September. The amount for October is x(t+1)*(0.05/12). This makes sense to me, but it doesn't work out. – Allison Sep 27 '16 at 07:29
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    The amount for October is x(t+1)*(0.05/12) No. The amount for October is x(t+1) by definition. You just need to calculate it based on the amount for the previous month $x(t+1) = x(t) (1+ ...)$ where $x(t) = 1000$. Fill in the blanks for whatever accumulates during the month. – dxiv Sep 27 '16 at 07:37
  • Okay. So then it would be x(t+1)=1000(1+0.05/12). That has to be it. I also hope that it is because I wrote down a similiar formula 800billion times without the 1+ portion and couldn't figure out why I wasn't getting any correct answer. – Allison Sep 27 '16 at 07:42

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It's just shifting the relation one step along. So instead of saying "Given x(t-1), write an expression for x(t)", it's saying "Given x(t), write an expression for x(t+1)". Equivalently, instead of "Given the balance last month, work out an expression for the balance this month" it's asking "Given the balance this month, work out an expression for the balance next month".

ConMan
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