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

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2524

2525

2424

2010

answer is B.

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Detailed Solution

It is given that

x^{2} - 5 x + 1 = 0

.
We can modify the given equation and write that,

x^{2} - 5 x + 1 = 0

\Rightarrow x^{2} + 1 = 5 x

⇒

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

⇒

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

\Rightarrow x^{2} + \frac{1}{x^{2}} = 23

Now,

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

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

\Rightarrow x^{3} + \frac{1}{x^{3}} = 110

We can write that,

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

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

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

\Rightarrow x^{5} + \frac{1}{x^{5}} = 2525

Hence, the correct option is 2.

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