$$\text { Show that } 3 ^ { n }

Name: \[\text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. }\]
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Question

$$\text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. }$$

Quizplus · Accepted Answer

The Answer is The basis step holds since

3 ^ { 7 } = 2187 < 5040 = 7 !

. For the inductive step assume that

3 ^ { k } < k !

where

k

is a positive integer greater than or equal to 7 . Using the inductive hypothesis we see that

3 ^ { k + 1 } = 3 \cdot 3 ^ { k } <

3 \cdot k ! < ( k + 1 ) k ! = ( k + 1 ) !

. This completes the proof.

Show that 3n<n ! whenever n is an integer with n≥7. \text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. } Show that 3n<n ! whenever n is an integer with n≥7.

$\text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. }$