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Question #e0122

1 Answer

Oct 5, 2015

Prove for $n = 7$ $n=7$
then prove that if true for $n = k - 1$ $n=k-1$ then it is true of $n = k$ $n=k$

Explanation:

Part 1
Using pencil and paper arithmetic
$XXX 3^{7} = 2187$ $color(white)("XXX")3^7= 2187$
$XXX 7! = 5040$ $color(white)("XXX")7! = 5040$
therefore if $n = 7$ $n=7$
$XXX 3^{n} < n!$ $color(white)("XXX")3^n < n!$

Part 2
Assume $3^{k - 1} < (k - 1)!$ $3^(k-1) < (k-1)!$
$\to$ $rarr$
$XXX 3 \cdot 3^{k - 1} < 3 \cdot [(k - 1)!]$ $color(white)("XXX")3*3^(k-1) < 3*[(k-1)!]$

$XXX 3^{k} < k \cdot [(k - 1)!] XXX$ $color(white)("XXX")3^k < k*[(k-1)!]color(white)("XXX")$ (since $k > 3$ $k > 3$ )

$XXX 3^{k} < k!$ $color(white)("XXX")3^k < k!$

QED

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