Question
Let n be a positive integer and $x>y$. Prove that $$\frac{x^n-y^n}{x-y}>ny^{n-1}$$ By choosing suitable values of x and y, further prove than $$\left(1+\frac{1}{n}\right)^n<\left(1+\frac{1}{n+1}\right)^{n+1}$$ and $$\left(1+\frac{1}{n}\right)^{n+1}>\left(1+\frac{1}{n+1}\right)^{n+2}$$
Answer
From this I can prove $$\frac{x^n-y^n}{x-y}>ny^{n-1}$$ By Using $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+...+y^{n-1})$$ I can get $$\frac{x^n-y^n}{x-y}=\frac{(x-y)(x^{n-1}+x^{n-2}y+...+y^{n-1})}{x-y}=x^{n-1}+x^{n-2}y+...+y^{n-1}>y^{n-1}+y^{n-2}*y+...+y^{n-1}=y^{n-1}+y^{n-1}+...+y^{n-1}=ny^{n-1}$$
But after this, I'm not sure what to do for the next question. Thank you in advance for the help
