The number of elements in the set $\{n \in \{1, 2, 3, \ldots, 100\} | (11)^n > (10)^n + (9)^n\}$ is…
The number of elements in the set $\{n \in \{1, 2, 3, \ldots, 100\} | (11)^n > (10)^n + (9)^n\}$ is ___________.
Solution
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 $(a + b)^n - (a - b)^n = 2 [C_1^n \cdot a^{n-1} \cdot b + C_3^n \cdot a^{n-3} \cdot b^3 + C_5^n \cdot a^{n-5} \cdot b^5 + \ldots]$
$\Rightarrow 2 [C_1^n \cdot 10^{n-1} + C_3^n \cdot 10^{n-3} + C_5^n \cdot 10^{n-5} + \ldots] > 10^n$
$\Rightarrow 2n \cdot 10^{n-1} + 2 [C_3^n \cdot 10^{n-3} + C_5^n \cdot 10^{n-5} + \ldots] > 10^n \ldots 1$
For $n = 5$, the LHS is $2 \cdot 5 \cdot 10^4 + 2 [C_3^5 \cdot 10^2 + C_5^5]$
$\Rightarrow 10^5 + 2 [C_3^5 \cdot 10^2 + C_5^5] > 10^5$ which is true.
Now, for $n = 6, 7, 8, \ldots, 100$
$2n \cdot 10^{n-1} > 10^n$
$\Rightarrow 2n \cdot 10^{n-1} + 2 [C_3^n \cdot 10^{n-3} + C_5^n \cdot 10^{n-5} + \ldots] > 10^n$
$\Rightarrow 11^n - 9^n > 10^n$ for all $n = 5, 6, 7, \ldots, 100$
For $n = 4$, the inequality 1 is not satisfied.
Hence, the inequality does not hold good for $n = 1, 2, 3, 4$
So, required number of elements $= 96$.
Asked in: JEE Main 2021 (22 Jul Shift 1)
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