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I was looking at the derivation for the constant $e$ when I came across this article.

Where does $e$ come from and what does it do?

Suppose you put £$1$ in a bank. The bank pays 4% interest a year, and this is credited to your account at the end of a year. A little thought shows that the end of five years and amount of money equal to £$(1+0.04)^5$ will sit in the bank (this bank charges no fees).

I am not understanding how this gives the amount in the bank after 5 years. It appears close but not exact.

My line of thinking is, after the first year you would have 0.04 of interest, then 0.08 after the second and so on until 0.20 after five years. This gives a total of 1.20 total in the bank after 5 years, but this equation gives 1.22 after this amount of time. Keep in mind, he mentions that this is not compounded interest.

Link to article: https://plus.maths.org/content/where-does-e-come-and-what-does-it-do-0

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    $\begingroup$ Interest is assumed to be compound. $\endgroup$ – Andrew Chin Mar 26 at 3:10
  • $\begingroup$ You earn interest on the interest, too, so long as you leave it on deposit. $\endgroup$ – saulspatz Mar 26 at 3:12
  • $\begingroup$ If it is not compound interest, it is wrong. For compound interest, you multiply by $1+i$ every year, which is what the formula does. For simple interest, you just add $iP$, where $P$ is the initial principal every year. $\endgroup$ – Ross Millikan Mar 26 at 3:12
  • $\begingroup$ I think this problem is nothing about math. $\endgroup$ – Zongxiang Yi Mar 26 at 3:13
  • $\begingroup$ Thank you for your replies, even your sarcastic one Zongxiang (how is this not math? Just because it’s not intense proofs? This shit makes people afraid to ask questions.). The reason I said it’s not compounded is in the next example he begins to say it’s compounded. I will edit to show you all the link. $\endgroup$ – ModularMan Mar 26 at 3:20
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Since it says the interest

... is credited to your account at the end of the year.

this means it is compounded interest, with the compounding period being $1$ year. Note that compounding means the interest is deposited, with interest then also being earned on the deposited interest (i.e., "compounded"). In the linked article, when the author talks about compounding more frequently, e.g., every quarter, month or even day, they didn't mean to imply their initial statement doesn't involve compounding.

Since there's interest being paid all of the previous year's interest as well, each next year's interest amount will increase slightly compared to the previous year's interest. Overall, as it indicates, the total amount would be

$$£(1 + 0.04)^5 \approx £1.22 \tag{1}\label{eq1A}$$

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  • $\begingroup$ Thank you, John. This was very helpful. $\endgroup$ – ModularMan Mar 26 at 3:24
  • $\begingroup$ @ModularMan You're welcome. Thanks for providing a link to the article. In that article, I hope you understand now that later on the author when the talks about compounding more frequently, e.g., every quarter, month or even day, but they didn't mean to imply their initial statement didn't involve compounding. $\endgroup$ – John Omielan Mar 26 at 3:25
  • $\begingroup$ Yes I see that now. It just appears that way when he goes from not saying it’s compounded to “however, if it was compounded” and gives the next example. $\endgroup$ – ModularMan Mar 26 at 3:28
  • $\begingroup$ @ModularMan I can see how that could be confusing. The author meant that if it were now being compounded every quarter instead of every year earlier, as opposed to now being compounded every quarter instead of not being compounded at all earlier. Also, note I have updated my answer to mention this more explicitly. $\endgroup$ – John Omielan Mar 26 at 3:31

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