The number of elements in the set {n in {1, 2 | vedclass

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(A) Given inequality: $11^{n} > 10^{n} + 9^{n}$Divide by $10^{n}$: $(1.1)^{n} > 1 + (0.9)^{n}$For $n=1$: $1.1 > 1 + 0.9 = 1.9$ (False)For $n=2$: $1.21

Vedclass · Accepted Answer

$96$

Vedclass · Answer

(A) Given inequality: $11^{n} > 10^{n} + 9^{n}$Divide by $10^{n}$: $(1.1)^{n} > 1 + (0.9)^{n}$For $n=1$: $1.1 > 1 + 0.9 = 1.9$ (False)For $n=2$: $1.21 > 1 + 0.81 = 1.81$ (False)For $n=3$: $1.331 > 1 + 0.729 = 1.729$ (False)For $n=4$: $1.4641 > 1 + 0.6561 = 1.6561$ (False)For $n=5$: $1.61051 > 1 + 0.59049 = 1.59049$ (True)For $n \ge 5$,the function $f(n) = (1.1)^{n} - (0.9)^{n}$ is strictly increasing.Since $f(5) > 1$,the inequality holds for all $n \in \{5, 6, 7, \ldots, 100\}$.The number of such elements is $100 - 5 + 1 = 96$.

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

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