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Question Number 119635 by Ar Brandon last updated on 25/Oct/20

$$\mathrm{If}\mathrm{a}\mathrm{function}f:\mathbb{R}\to \mathbb{R}\mathrm{satisfies}\mathrm{the}\mathrm{relation}$$$$f(\mathrm{x}+1)+f(\mathrm{x}-1)=\sqrt{3}f\left(\mathrm{x}\right)\mathrm{for}\mathrm{all}\mathrm{x}\in \mathbb{R}$$$$\mathrm{then}\mathrm{a}\mathrm{period}\mathrm{of}f\mathrm{is}$$$$\left(\mathrm{A}\right)10\left(\mathrm{B}\right)12\left(\mathrm{C}\right)6\left(\mathrm{D}\right)4$$

Answered by Olaf last updated on 26/Oct/20

$$f(x+1)+f(x-1)=\sqrt{3}f\left(x\right)$$$$$$$$f(x+2)+f\left(x\right)=\sqrt{3}f(x+1)$$$$f(x+2)+f\left(x\right)=\sqrt{3}[\sqrt{3}f\left(x\right)-f(x-1)]$$$$f(x+2)=2f\left(x\right)-\sqrt{3}f(x-1)$$$$$$$$f(x+3)=2f(x+1)-\sqrt{3}f\left(x\right)$$$$f(x+3)=2[\sqrt{3}f\left(x\right)-f(x-1)]-\sqrt{3}f\left(x\right)$$$$f(x+3)=\sqrt{3}f\left(x\right)-2f(x-1)$$$$$$$$f(x+4)=\sqrt{3}f(x+1)-2f\left(x\right)$$$$f(x+4)=\sqrt{3}[\sqrt{3}f\left(x\right)-f(x-1)]-2f\left(x\right)$$$$f(x+4)=f\left(x\right)-\sqrt{3}f(x-1)$$$$$$$$f(x+5)=f(x+1)-\sqrt{3}f\left(x\right)$$$$f(x+5)=f(x+1)-[f(x+1)+f(x-1)]$$$$f(x+5)=-f(x-1)$$$$$$$$f(x+6)=-f\left(x\right)$$$$\Rightarrow f(x+12)=-f(x+6)$$$$f(x+12)=-[-f\left(x\right)]$$$$f(x+12)=f\left(x\right)$$$$$$$$\mathrm{Period}\mathrm{is}12.$$

Commented by Ar Brandon last updated on 26/Oct/20

Thank you Sir

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