I want to solve these two simultaneous equations.
$\frac{1-1.2^{-n}}{0.2}x=295 \rightarrow*\\ \frac{1-1.15^{-n}}{0.15}x=386 \rightarrow**$
My attempt first I thought to divide *, by** but then it is hard to simplify.
Then i tried expanding $\frac{1-1.2^{-n}}{0.2}=1+1.2^{-1}+\cdots+1.2^{-n}$ but it also didn't make any progress. Is there a direct way to solve this equations?
Dividing is a good idea, you get $$-11-579\times1.2^n+590\times 1.15^n=0$$
For $n$ integer the only solution is $n=0$, but this is not a solution of original equation.
Only way is to have $n$ real, in this case the derivative is easier to study (since the constant term disappear) and you can show it has only $1$ other solution.
Although it seems to me you can only find a numerical solution for this $n\approx -25.6310417$ and consequently $x\approx 59.55645625$.