Question

If $x^2-5x+1=0$, then $\frac{x^{10}+1}{x^5}$ has the value

• A. $2524$
• B. $2525$
• C. $2424$
• D. $2010$

Answer 1

The correct option is B $2525$

$x^2-5x+1=0$

$\therefore x^2+1=5x$

$\therefore x+\frac{1}{x}=5$

$\therefore x^2+\frac{1}{x^2}=25-2=23$

$\therefore x^3+\frac{1}{x^3}={(x+\frac{1}{x})}^3-3(x+\frac{1}{x})$

$\therefore x^3+\frac{1}{x^3}=125-15$

$\therefore x^3+\frac{1}{x^3}=110$

Now $(x^3+\frac{1}{x^2})(x^3+\frac{1}{x^3})=56\times 23$

$\therefore x^5+x+\frac{1}{x}+\frac{1}{x^5}=110\times 23$

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

$\therefore x^5+\frac{1}{x^5}=2525$.