Lucas Numbers?

Algebra Level 4

$\sqrt[8]{2207-\dfrac{1}{2207-\dfrac{1}{2207-\cdots}}}$

The express above can be expressed as $\dfrac{a+b\sqrt{c}}{d}$ , where $a,b,c,d\in \Bbb{Z}$ with $\gcd(a, b, d) = 1$ and $c$ is a square free integer. Find the value of $a + b + c + d$ .

The answer is 11.

1 solution

Danish Ahmed
Feb 26, 2015

Let $x$ be the answer.

Then, $x^8 + \dfrac {1}{x^8} = 2207$ .Then, we see that $x^4 + \dfrac {1}{x^4} = 47$ , so $x^2 + \dfrac {1}{x^2} = 7$ , so $x + \dfrac {1}{x} = 3$ . Thus, $x = \dfrac {3+\sqrt{5}}{2}$

So $a+b+c+d = 11$ .

Generalization :

Let $L_n$ be the nth Lucas number.

Then, $\sqrt[n]{L_{2n} - \dfrac {1}{L_{2n} - \dfrac {1}{L_{2n} - \cdots}}} = \dfrac {3+\sqrt{5}}{2}$ .

Lucas Numbers?

The answer is 11.

1 solution

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