find the interest rate in the form of compound interest with recurring contribution

Question

I have the following formula to find the final value of compound interest with recurring contribution:

Where:
S: is final amount
T: my recurring value
i: interest rate
n: recurrence period

what I want is to isolate i in this formula

Answer 1

Comment: If you mean i in term of S, T and n then let $(1+i)^n=A$, you may write :

$1+i=A^{\frac 1n}$

$$S=T\cdot \frac {A-1}{A^{\frac 1n}-1}$$

$$S\cdot A^{\frac 1n}-T\cdot A=S-T$$

That is if interest rate is 0 then $A^{\frac 1n}=A=1$

You may rewrite this as:

$$S (1+i)-T(1+i)^n=S-T$$