TITLE: How to derive the formula for the amount?
QUESTION [0 upvotes]: Compounded half-yearly:
Amount after half year = $P(1+\frac {R/2}{100})^{2n}$

How can we derive this formula?

REPLY [1 votes]: Let the half-yearly compound interest rate be $I$, in percent.
After the first half year, the amount is $P+\frac I{100}P=(1+\frac I{100})P$.
After the second half year, $(1+\frac I{100})P+\frac I{100}(1+\frac I{100})P=(1+\frac I{100})^2P$, and the general pattern is obvious.

Now we have a little technicality, as what is given is a yearly interest rate, $R$, which is a simple rate instead of a compound one. That means that if after a year you get $R\%$, after a half year you get $\frac R2\%$, and this is our $I$.
Note that mathematician's logic would be that $R$ also be a compound rate, and we should have $(1+\frac I{100})^2=1+\frac R{100}$ or $I=100(\sqrt{1+\frac R{100}}-1)$, but this is not the way it is.
{"set_name": "stack_exchange", "score": 0, "question_id": 3845902}