Answer
$n=-\dfrac{\ln (1-\dfrac{P_i}{A})}{\ln (1+i)}$
Work Step by Step
The given expression can be written as:
$1-(1+i)^{-n}=\dfrac{P_i}{A}$
or,$(1+i)^{-n}=1-\dfrac{P_i}{A}$
We need to take natural log of both sides
$ \ln (1+i)^{-n}=\ln (1-\dfrac{P_i}{A})$
Now, apply logarithmic property : $\log a^ b=b \log a$ .
$-n \ln (1+i)=\ln (1-\dfrac{P_i}{A})$
Now, we will solve for $n$.
Therefore, our answer is: $n=-\dfrac{\ln (1-\dfrac{P_i}{A})}{\ln (1+i)}$
