The number of elements in the set n1,2,,100:(11)^n(10)^n+(9)^n is Answer: 96

Question

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

Answer: 96

Accepted Answer

$11^n-9^n>10^n\Rightarrow(10+1)^n-(10-1)^n>10^n$

$$2\left[{}^nC_1\,10^{n-1}+{}^nC_3\,10^{n-3}+\cdots\right]>10^n$$

Checking small $n$: for $n=1,2,3,4$ this fails, but from $n=5$ onward the inequality holds (verified via the leading terms dominating). So valid $n\in\{5,6,\dots,100\}$: 96 elements.

Solutions · 1