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

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Basic Mathematics

The number of elements in the set

{n \in {1, 2, 3, \dots, 100} ∣ (11)^{n} > (10)^{n} + (9)^{n}}

is ____

Answer: 96

Dividing by

1 0^{n}

, the condition becomes

(\frac{11}{10})^{n} - (\frac{9}{10})^{n} > 1

The left side increases with

n

, since

(1.1)^{n}

grows while

(0.9)^{n}

decreases, so the integers satisfying the inequality form a tail of

{1, 2, \dots, 100}

. At

n = 4

1 1^{4} = 14641

against

1 0^{4} + 9^{4} = 16561

, so the inequality fails; at

n = 5

1 1^{5} = 161051

against

1 0^{5} + 9^{5} = 159049

, so it holds. The inequality therefore holds exactly for

5 \leq n \leq 100

, and the set has

96

elements.

JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Integer answer