Question

Alguien me ayuda a hallar el valor de T?

Answer 1

$T=\sqrt[n-1]{\frac{x^{n-1}+1}{x^{1-n}+1}}+\sqrt[n-2]{\frac{x^{n-2}+1}{x^{2-n}+1}}+\sqrt[n-3]{\frac{x^{n-3}+1}{x^{3-n}+1}}\\ \\T=\left(\frac{x^{n-1}+1}{x^{1-n}+1}\right)^{\frac{1}{n-1}}+\left(\frac{x^{n-2}+1}{x^{2-n}+1}\right)^{\frac{1}{n-2}}+\left(\frac{x^{n-3}+1}{x^{3-n}+1}\right)^{\frac{1}{n-3}}$

$T=\left(\frac{x^{n-1}+1}{(x^{1-n})(1+x^{n-1})}\right)^{\frac{1}{n-1}}+\left(\frac{x^{n-2}+1}{(x^{2-n})(1+x^{n-2})}\right)^{\frac{1}{n-2}}+\left(\frac{x^{n-3}+1}{(x^{3-n})(1+x^{n+3})}\right)^{\frac{1}{n-3}}\\ \\T=\left(\frac{1}{x^{1-n}}\right)^{\frac{1}{n-1}}+\left(\frac{1}{x^{2-n}}\right)^{\frac{1}{n-2}}+\left(\frac{1}{x^{3-n}}\right)^{\frac{1}{n-3}}\\ \\T=(x^{n-1})^{\frac{1}{n-1}}+(x^{n-2})^{\frac{1}{n-2}}+(x^{n-3})^{\frac{1}{n-3}}\\ \\T=x+x+x\\T=\textbf{3x}$