Q101 / 180 · KCET 2020 · Mathematics

If $P(n): 2^{n} < n$ ! Then the smallest positive integer for which $P(n)$ is true if

  • A
    2
  • B
    3
  • Correct answer
    C
    4
  • D
    5

Solution

We have, $\begin{aligned} &P(n)=2^{n} < n ! \\ &P(1)=2 < 1 ! \text { False } \end{aligned}$ $\begin{aligned} &P(2)=2^{2} < 2 ! \text { False } \\ &P(3)=2^{3} < 3 \text { ! False } \\ &P(4)=2^{4} < 4 \text { ! True } \end{aligned}$ $\therefore$ The smallest position integer for which $P(n)$ is true for $n=4$.

Hence, the correct answer is option: 'C'.

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