What Happened to Christmas?

Algebra Level 3

If

12 + 25 12 + 25 12 + 25 ⋯ = a + b c , \sqrt{12 + 25\sqrt{12 + 25\sqrt{12 + 25\cdots}}} = \frac{a + \sqrt{b}}{c},

for positive integers a , b , a, b, and c c such that b b is not divisible by the square of any prime, find a + b + c . a+b+c.


The answer is 700.

Steven Yuan by Steven Yuan , 20, USA

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1 solution

Steven Yuan
Dec 26, 2014

Let x = 12 + 25 12 + 25 12 + 25 ⋯ . x = \sqrt{12 + 25\sqrt{12 + 25\sqrt{12 + 25\cdots}}}. We have

x = 12 + 25 x x 2 = 12 + 25 x x 2 − 25 x − 12 = 0 x = 25 + 625 + 48 2 x = 25 + 673 2 \begin{aligned} x &= \sqrt{12 + 25x} \\ x^2 &= 12 + 25x \\ x^2 - 25x - 12 &= 0 \\ x &= \frac{25 + \sqrt{625 + 48}}{2} \\ x &= \frac{25 + \sqrt{673}}{2} \\ \end{aligned}

Note that we discard the negative root because our infinite radical is positive. Now, all that's left is to figure out whether the radical does converge to the value we found. Using a calculator, we find it indeed does, so a + b + c = 25 + 673 + 2 = 700 . a + b + c = 25 + 673 + 2 = \boxed{700}.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

did the same way . thanks i didnt know why to ignore the negative root

Guru Prasaadh - 6 years, 5 months ago

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