What should I substitute?

Algebra Level 4

if f ( x ) f(x) satisfies the condition f ( x − 1 ) + f ( x + 1 ) = 3 f ( x ) f(x-1)+f(x+1)=\sqrt{3}f(x) then it is periodic with period T T . Find the value of T T .

None of these choices 12 2 6 10

Sabhrant Sachan by Sabhrant Sachan , 21, India

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1 solution

Otto Bretscher
May 11, 2016

f ( x ) f(x) satisfies the recursive equation of U n ( 3 2 ) = U n ( cos ⁡ ( π 6 ) ) = 2 sin ⁡ ( π 6 ( n + 1 ) ) U_n(\frac{\sqrt{3}}{2})=U_n\left(\cos(\frac{\pi}{6})\right)=2\sin(\frac{\pi}{6}(n+1)) , where U n ( x ) U_n(x) is the Chebyshev polynomial of the second kind, so that a period is 2 π π / 6 = 12 \frac{2\pi}{\pi/6}=\boxed{12}

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