Describe the relationship between the mathematics of calculating simple interest, and the mathematics behind analyzing arithmetic sequences. How might simple interest be described as a problem of applied arithmetic sequences?
This question completely took me by surprise. i dont understand how arithmetic sequences can be related to simple interest. i am confused and lost. please help.
Thanks, i am in grade 11, if it is of any help.
An arithmetic progression is a sequence of numbers where the difference between consecutive terms is constant: e.g. the sequence $10, 12, 14, 16 ...$ is an arithmetic progression because the difference between consecutive terms is $2$. This is exactly the type of sequence you see when looking at how a debt grows at regular intervals with simple interest: the first term is the principal (the "initial debt") and each subsequent term is equal to the previous one plus the interest due on the principal for the the last period. In the example above, the initial debt would be $10$, and the interest would be $20\%$ per period (so, $2$ per period).
Compare this with a geometric progression, a sequence of numbers where the ratio (instead of the difference) between consecutive terms is constant. E.g. $10, 12, 14.4, 17.28 ...$ is a geometric progression, because the ratio between consecutive terms is $1.2$. This is exactly the type of sequence you see when looking at how a debt grows at regular intervals with compound interest, i.e. when interest, once accrued, generates further interest by itself. In the example above, you still have a $20\%$ interest rate, but after the first period (when your debt grows from $10$ to $12$) those extra $2$ units of debt you accrued start accruing their own interest, so after the second period you owe $10$ (the principal) plus $2\times 2$ (the interest on the principal for the two time periods, like with simple interest) plus $0.4$, which is the interest accrued during the second period on the interest accrued during the first ($20\%$ of $20\%$ of $10$) for a total of $14.4$. With compound interest then, at each time period, your new debt becomes ($100\%$ plus interest rate) $\times$ the previous debt, hence the geometric progression.