Prove that n 3n for each integer n greaterthanorequalto 7So

Prove that n! > 3^n for each integer n greaterthanorequalto 7.

Solution

Let us prove this by induction.

Let n =7

7! = 5040 and 3^7 = 2187

So n! > 3^n when n =7

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Let us assume k! >3^k for some k >=7

Consider k+1

(k+1)! = k!(k+1) > 3^k(k+1)

Since k>7, k+1 >3

So (k+1)! = k!(k+1) > 3^k(k+1)>3^k(3)

Or (k+1)! >3^k+1, if true for n =k

Already true for n =7

Hence true for n =8,9,10,.....

Proved by mathematical induction for n >=7