Joel's Problem 3: All good... with exceptions

Algebra Level 5

Consider the infinite sequence of real numbers x 1 , x 2 , x 3 x_{1}, x_{2}, x_{3} ... such that for all positive integers n n , x n + 1 = ( 2 − 3 ) + x n 1 − ( 2 − 3 ) x n . x_{n+1}=\frac {(2-\sqrt{3})+x_{n}}{1-(2-\sqrt {3})x_{n}}.

For some values of x 1 x_{1} , there would exist a term of the sequence that is undefined. How many such values are there?


The answer is 11.

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Joel Tan by Joel Tan , 21, Singapore

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1 solution

Abhishek Sinha
Aug 27, 2014

Taking tan ⁡ − 1 \tan^{-1} of both sides, we have tan ⁡ − 1 x n + 1 = tan ⁡ − 1 ( x n ) + π 12 \tan^{-1}x_{n+1}=\tan^{-1}(x_n )+ \frac{\pi}{12} , where we consider the principal values only. For some value of tan ⁡ − 1 x 1 \tan^{-1}x_1 , a term of the sequence will be undefined when we have tan ⁡ − 1 x n + 1 = π 2 \tan^{-1}x_{n+1}=\frac{\pi}{2} for some n n and clearly there are 11 such values given by − π 2 + k π 12 , k = 1 , 2 , … , 11 -\frac{\pi}{2}+k\frac{\pi}{12}, k=1,2,\ldots, 11 .

11 Helpful 0 Interesting 0 Brilliant 0 Confused

Respected Abhishek Sinha sir, please guide me to improve my skills in electrical side.

Krv Ramanan - 6 years, 9 months ago

I guess the main mistake in this problem would be including x=tan 90°

Joel Tan - 6 years, 9 months ago

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