Question

The number of elements in the set $\{n\in \{1,2,3,...,100\}|(11{)}^n>(10{)}^n+(9{)}^n\}$ is

Answer 1

${11}^n-9^n>{10}^n$
$\Rightarrow {(1+\frac{1}{10})}^n-{(1-\frac{1}{10})}^n>1$
$\begin{array}{c}\Rightarrow {}^n{C}_1.\frac{1}{10}+\underset{}{{}^n{C}_3.\frac{1}{{10}^3}+{}^n{C}_5.\frac{1}{{10}^5}+...}>\frac{1}{2}\\ \text{Neglecting these terms}\end{array}$

$\Rightarrow n\ge 5$

Possible values of $n=5,6,7,8,...,100$
Clealy, it has $96$ elements.