Help!! How prove it??

Question

#(a+b)^n + (a+1/b)^n >= 2^(n+1) * a^(n/2)#

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Answer 1 · 2017-12-02T20:54:16

See below.

#(a+b)^n + (a+1/b)^n >= 2 sqrt(((a+b)(a+1/b))^n)#

but

#a+b ge 2 sqrt(ab)#

and

#a+1/b ge 2 sqrt(a/b)#

so

#2 sqrt(((a+b)(a+1/b))^n) ge 2 sqrt((2^2sqrt(ab)sqrt(a/b))^n) = 2 sqrt((2^2a)^n) = 2^(n+1)a^(n/2)#

and finally

#(a+b)^n + (a+1/b)^n >=2^(n+1)a^(n/2)#