Mathematics - Binomial Theorem Question with Solution | TestHub

We have to check ${11}^n>{10}^n+9^n$

$\Rightarrow {11}^n-9^n>{10}^n$

$\Rightarrow (10+1{)}^n-(10-1{)}^n>{10}^n$

Using ${\left(a+b\right)}^n-{\left(a-b\right)}^n=2\left[{}^{n}C_1·a^{n-1}·b+{}^{n}C_3·a^{n-3}·b^3+{}^{n}C_5·a^{n-5}·b^5+...\right]$

$\Rightarrow 2\left[{}^{n}C_1·{10}^{n-1}+{}^{n}C_3·{10}^{n-3}+{}^{n}C_5·{10}^{n-5}+\cdots \cdots \right]>{10}^n$

$\Rightarrow 2n·{10}^{n-1}+2\left[{}^{n}C_3·{10}^{n-3}+{}^{n}C_5·{10}^{n-5}+\cdots \cdots \right]>{10}^n$ $\dots \left(1\right)$

For $n=5,$ the $L.H.S.$ is $2·5·{10}^4+2\left[{}^{5}C_3·{10}^2+{}^{5}C_5\right]$

$\Rightarrow {10}^5+2\left[{}^{5}C_3·{10}^2+{}^{5}C_5\right]>{10}^5$ which is true.

Now, for $n=6, 7, 8, \dots \dots , 100$

$2n·{10}^{n-1}>{10}^n$

$\Rightarrow 2n·{10}^{n-1}+2\left[{}^{n}C_3·{10}^{n-3}+{}^{n}C_5·{10}^{n-5}+\cdots \cdots \right]>{10}^n$

$\Rightarrow {11}^n-9^n>{10}^n$ for all $n=5, 6, 7, \dots \dots , 100$

For $n=4,$ the inequality $\left(1\right)$ is not satisfied.

Hence, the inequality does not hold good for $n=1, 2, 3, 4$

So, required number of elements$=96.$