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
Since it says the interest
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
$$£(1 + 0.04)^5 \approx £1.22 \tag{1}\label{eq1A}$$