Question

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

Answer 1

$11^{n}>10^{n}+9^{n}$
$\Rightarrow 11^{n}-9^{n}>10^{n}$
$\Rightarrow(10+1)^{n}-(10-1)^{n}>10^{n}$
$\Rightarrow\left\{{ }^{n} C_{1} \cdot 10^{n-1}+{ }^{n} C_{3} 10^{n-0}+{ }^{n} C_{5} 10^{n-5}+\ldots \ldots\right\}>10^{n}$
$\Rightarrow 2 n \cdot 10^{n-1}+2\left\{{ }^{n} C_{3} 10^{n-3}+{ }^{n} C_{5} 10^{n-5}+\ldots \ldots .\right\}>10^{n}$ .... (1)
For $n=5 $
$10^{5}+2\left\{{ }^{5} C_{3} 10^{2}+{ }^{5} C_{5}\right\}>10^{5}$ (True)
For $ n=6,7,8, \ldots \ldots .100$
$2 n 10^{n-1} > 10^{n} $
$\Rightarrow 2 n 10^{n-1}+2\left\{{ }^{n} C_{3} 10^{n-3}+{ }^{n} C_{5} 10^{n-5}+\ldots \ldots . .3 > 10^{\circ}\right. $
$\Rightarrow 11^{n}-9^{n} > 10^{n} $ For $ n=5,6,7, \ldots \ldots .100$
For $ n=4, $ Inequality (1) is not satisfied
$\Rightarrow $ Inequality does not hold good for
$ N=1,2,3,4$
So, required number of elements $=96$