Derive Compound Interest Formula

Question

I'm trying to derive Compound Interest Formula

$100 is given 1% interest per month

I can manually work this by hand (and Excel)

See how the formula just expands?

How to create simple formula such that person can find total interest paid at a given time?

Screenshot: Written Long Hand (typed and formula in Excel)

Screenshot: Interest paid each month, plus total for year

Answer 1

If you start with $P$ and the interest rate is $r$ each period compounding over $n$ periods then you should end up with $$P\times (1+r)^n$$

So as an example in Excel with $P=100$ and $r=1\%$ and $n=12$ you can write this as =100*(1+1%)^12 which should give you about $112.682503013197$.

Subtract $100$ from this to show you have $12.682503013197$ more than you started with, essentially the same result as your calculation.

Answer 2

It is commonly known that the compound interest formula is $$T_n=P\left(1+r\right)^n$$where $T_n$ is the total amount after $n$ iterations of interest being added on,
$P$ is the principal value that is originally invested,
$r$ is the interest rate as a decimal ($0.01$ in this case),
$n$ is the number of iterations for which you wish to compute the end value.

Answer 3

The value after the $n$th month is given by: $$100*(1.01)^n-100$$ This is because the total value after $n$ months is $((100 * 1.01)*1.01)*1.01...=100*(1.01)^n$ and we want the difference from the original value - so subtract 100.