Question

Simplify the following and express with positive index:
$[1-{\{1-(1-n{)}^{-1}\}}^{-1}{]}^{-1}$

Answer 1

Simplifying the given expression
with positive index
$[1-{\{1-(1-n{)}^{-1}\}}^{-1}{]}^{-1}$
$=[1-{\{1-(1-n{)}^{-1}\}}^{-1}{]}^{-1}$
$={[1-{\{1-\frac{1}{1-n}\}}^{-1}]}^{-1}$ $[\because a^{-1=\frac{1}{a}}]$
$={[1-{\left\{\frac{1-n-1}{1-n}\right\}}^{-1}]}^{-1}$
$={[1-\frac{1}{\frac{-n}{1-n}}]}^{-1}$ $[\because a^{-1=\frac{1}{a}}]$
On further calculation, we have:
${[1+\frac{1-n}{n}]}^{-1}={\left[\frac{n+1-n}{n}\right]}^{-1}={\left[\frac{1}{n}\right]}^{-1}=n$

Hence, the simplified value with positive index is n.