I'm trying to calculate the interest rate in compound interest, but I find that the online formula can't get the answer I want. I don't know why. Does anyone know what the specific formula is? I know the calculator can done that, but I need to use formula for programming.
The formula I use is r = ( FV / PV )1/n - 1, but the result is 7.56 something, but not as what the picture shows.
We have the following formula for calculating the future value ($FV$) of a deposit ($PV$) with an annual interest rate of $r\%$ compounded $m$ times per year for a total of $n$ times: $$FV = PV\left(1 + \frac{r}{m}\right)^n.$$ In the example you linked we are given the following information: $PV = 5000$, $FV = 6000$, $m = 12$ and $n = 2.5 \times 12 = 30$. All that we need to find is $r$, the annual interest rate.
Plugging the known values into the formula: $$6000 = 5000 \left( 1 + \frac{r}{12} \right)^{30}.$$ We need to solve for $r$ by rearranging this equation: \begin{align} r &= 12 \left( \sqrt[30]{\frac{6000}{5000}} - 1 \right) \\ &= 0.07315\ldots. \end{align} So the annual interest rate is approximately $7.32\%$.
The general formula for annual interest $r$ (which you can get by rearranging the original compound interest formula) is $$r = m \left( \sqrt[n]{\frac{FV}{PV}} - 1\right).$$ Note that taking the $n$-th root of $FV/PV$ is the same as raising $FV/PV$ to the power of $1/n$.