# #y_n=log x_n, n =2,3,4,...and y_n-(n-1)/n y_(n-1)=1/n log n#, with #y_2=log sqrt2#, how do you prove that #x_n=(n!)^(1/n)#?

##### 1 Answer

By induction

#### Explanation:

Note that as

** Proof:** (By induction)

*Base Case:* For

*Inductive Hypothesis:* Suppose that

*Induction Step:* We wish to show that

#=1/(k+1)[log(k+1)+klog(x_k)]#

#=1/(k+1)[log(k+1)+klog((k!)^(1/k))]#

#=1/(k+1)[log(k+1)+log(k!)]#

#=1/(k+1)log(k!(k+1))#

#=1/(k+1)log((k+1)!)#

#=log([(k+1)!]^(1/(k+1)))#

meaning

We have supposed true for

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