Interest on interest

Put one hundred tokens in a savings box that pays ten per cent a year. After one year you have one hundred and ten.

$100$ tokens at $10\%$ a year: after one year $100\times 1.1=110$.

Money after $n$ years: $A_n=100\cdot 1.1^n$, so $A_1=110$.

In the second year the ten per cent is paid on one hundred and ten, not on one hundred: that is eleven more, so you have one hundred and twenty-one. In the third year you get twelve and a tenth, and you have 133.1.

Year two pays on $110$: $110\times 1.1=121$. Year three: $121\times 1.1=133.1$, that is $100\times 1.1^3=133.1$.

$A_2=121$ and $A_3=133.1$.

If interest were paid only on the first hundred, you would get ten a year and have one hundred and thirty after three years. The extra 3.1 is interest on interest.

Simple interest would give $100+3\times 10=130$. The extra $133.1-130=3.1$ is interest on interest.

Simple interest gives $130$; $A_3-130=3.1$.

A rule of thumb: divide seventy-two by the rate to find the years to double, here seven and two tenths. In truth it lies between seven and eight years: after seven years you have not yet doubled, and after eight you have.

The rule of seventy-two: $\frac{72}{10}=7.2$ years. In truth $1.1^7\approx 1.95<2<1.1^8\approx 2.14$.

$\frac{72}{10}=7.2$ years; truly $A_7<200<A_8$.