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  • The number of elements in the set <img alt="\{n \in\{1,2,3, \ldots ., 100\}: \left.(11)^{\mathrm{n}}>(10)^{\mathrm{n}}+(9)^{\mathrm{n}}\right\}" src="https://entrancecorner.codecogs.com/gif.latex?%5C%7Bn%20%5Cin%5C%7B1%2C2%2C3%2C%20%5Cldots%20.%2C%2

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

Answers (1)

best_answer

\begin{aligned} & 11^{n}>10^{n}+9^{n} \\ \Rightarrow & 11^{n}-9^{n}>10^{n} \\ \Rightarrow &(10+1)^{n}-(10-1)^{n}>10^{n} \\ \Rightarrow & {\left[10^{n}+{ }^{n} C_{1} \cdot 10^{n-1}+{ }^{10} C_{2} \cdot 10^{n-2}+\cdots \right.} \end{aligned}

\begin{gathered} \left.-\left(10^{n}-{ }^{n} C_{1} 10^{n-1}+{ }^{10} C_{2} 10^{n-2}-\cdots\right)\right]>10^{n} \\ \Rightarrow 2\left({ }^{n} C_{1} \cdot 10^{n-1}+{ }^{n} C_{3} 10^{n-3}+\cdots \cdot .\right)>10^{n}-(i) \end{gathered}

By hit and trial, n= 4 does not satisfy it and n= 5 satisfy it.

Also 11^{n} increases at faster rate than 10^{n}+9^{n}

So n= 5,6,7.........100 satisfy it.

\text{Total number}= 96

Hence, the correct answer is (96).

Posted by

Kuldeep Maurya

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