Start with
[tex]18020=120\left(1-\dfrac{(1+0.4233)^{-n}}{0.4233}\right)[/tex]
Divide both sides by 120:
[tex]\dfrac{18020}{120}=1-\dfrac{(1+0.4233)^{-n}}{0.4233}[/tex]
Simplify the dfraction on the left hand side, and the sum in the parenthesis of the right hand side:
[tex]\dfrac{901}{6}=1-\dfrac{(1.4233)^{-n}}{0.4233}[/tex]
Subtract 1 from both sides:
[tex]\dfrac{901}{6}-1=-\dfrac{(1.4233)^{-n}}{0.4233}[/tex]
Change side to both sides:
[tex]1-\dfrac{901}{6}=\dfrac{(1.4233)^{-n}}{0.4233}[/tex]
Simplify the left hand side:
[tex]-\dfrac{895}{6}=\dfrac{(1.4233)^{-n}}{0.4233}[/tex]
Multiply both sides by 0.4233:
[tex]-0.4233\cdot\dfrac{895}{6}=(1.4233)^{-n}[/tex]
Simplify the left hand side:
[tex]-\dfrac{378.8535}{6}=(1.4233)^{-n}[/tex]
Apply the definition of negative exponents:
[tex]-\dfrac{378.8535}{6}=\dfrac{1}{1.4233^n}[/tex]
Invert both sides:
[tex]-\dfrac{6}{378.8535}=1.4233^n[/tex]
This is why the equation is impossible: the function [tex]f(x) = 1.4233^x[/tex] is an exponential function, and as such, it is always positive. So, there cannot be a value for [tex]n[/tex] such that the last equation is satisfied.
