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  4. If P (n): 2n < n! Then the smallest positive integer for which P (n) is true, is

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Q. If $P (n) : 2^n < n!$ Then the smallest positive integer for which P (n) is true, is

KCETKCET 2019Principle of Mathematical Induction

Solution:

$P\left(n\right) ; 2^{n} <\, n ! $
$n=2; 2^{2}=4, n !=2 !=2$
$4≮ 2$
$n=3; 2^{3}=8, 3!=6$
$8≮ 6$
$n=4; 2^{4}=16, 4 !=24$
$16 <\, 24$
$\therefore \, n=4 $