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Can somebody show these inequalities by induction:

  1. $2^n > 2n+1,$ where $ n \geqslant 3.$
  2. $ 2^{n-1}(x^n+y^n) > (x+y)^n,$ where $ x+y>0, x \neq y$ and $ n \geqslant 1.$

1 Answers 1

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Hint for 1: The base case is simple enough, just observe $2^3=8>7=2\cdot 3+1$. For the inductive case, note that if $2^n>2n+1$, what happens when you multiply both sides by $2$?

Hint for 2: To start, this should be $2^{n-1}(x^n+y^n)\geq (x+y)^n$ rather than $2^{n-1}(x^n+y^n)> (x+y)^n$ as if $n=1$ the two sides are equal. This gives you the base case. If $2^{n-1}(x^n+y^n)\geq (x+y)^n$, what happens when you multiply both sides by $(x+y)$?