If $$x+\frac{1}{x}=5$$, then what is the value of $$x^5+\frac{1}{x^5}$$ ?

Given : $$x+\frac{1}{x}=5$$ ------------(i)

Squaring both sides,

=> $$(x+\frac{1}{x})^2=(5)^2$$

=> $$x^2+\frac{1}{x^2}+2(x)(\frac{1}{x})=25$$

=> $$x^2+\frac{1}{x^2}=25-2=23$$ ----------(ii)

Now, cubing equation (i), we get :

=> $$(x+\frac{1}{x})^3=(5)^3$$

=> $$x^3+\frac{1}{x^3}+3(x)(\frac{1}{x})(x+\frac{1}{x})=125$$

=> $$x^3+\frac{1}{x^3}+3(5)=125$$

=> $$x^3+\frac{1}{x^3}=125-15=110$$ -----------(iii)

Multiplying equations (ii) and (iii),

=> $$(x^2+\frac{1}{x^2})(x^3+\frac{1}{x^3})=23\times110$$

=> $$x^5+\frac{1}{x^5}+x+\frac{1}{x}=2530$$

=> $$x^5+\frac{1}{x^5}+5=2530$$

=> $$x^5+\frac{1}{x^5}=2530-5=2525$$ 

=> Ans - (B)