First slide

Introduction to P.M.I

Question

The inequality $n! > 2^{n - 1}$ is true for

Moderate

A

n>2
B

$n \in N$
C

n > 3
D

None of these

Solution

$\begin{array}{l} Let P (n) \equiv n! > 2^{n - 1} \\ P (3) \equiv 6 > 4 \\ Let P (k) \equiv k! > 2^{k - 1} is true. \\ P (k + 1) = (k + 1)! = (k + 1) k! \\ > (k + 1) 2^{k - 1} \\ > 2^{k} (as k + 1 > 2) \end{array}$

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