The number of elements in the set {n $\in$ {1, 2, 3, ......., 100} | (11)n > (10)n + (9)n} is ______________.
Answer (integer)
96
Solution
${11^n} > {10^n} + {9^n}$<br><br>$\Rightarrow {11^n} - {9^n} > {10^n}$<br><br>$\Rightarrow {(10 + 1)^n} - {(10 - 1)^n} > {10^n}$<br><br>$$ \Rightarrow 2\{ {}^n{C_1}{.10^{n - 1}} + {}^n{C_3}{10^{n - 10}} + {}^n{C_5}{10^{n - 5}} + .....\} > {10^n}$$
<br><br>$\Rightarrow$ $${1 \over 5}\left[ {{}^n{C_1}{{10}^n} + {}^n{C_3}{{10}^{n - 2}} + {}^n{C_5}{{10}^{n - 4}} + .....} \right] > {10^n}$$
<br><br>$\Rightarrow$ $${1 \over 5}\left[ {{}^n{C_1} + {}^n{C_3}{{10}^{ - 2}} + {}^n{C_5}{{10}^{ - 4}} + .....} \right] > 1$$
<br><br>Clearly the above inequality is true for n $\ge$ 5
<br><br>For n = 4, we have $${1 \over 5}\left[ {4 + {4 \over {{{10}^2}}}} \right] = {4 \over 5}\left( {{{101} \over {100}}} \right) < 1$$
<br><br>$\Rightarrow$ Inequality does not hold good for n = 1, 2, 3, 4
<br><br>So, required number of elements ={5, 6, 7, ......., 100} = 96
About this question
Subject: Mathematics · Chapter: Binomial Theorem · Topic: Binomial Expansion
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