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Question-9847




Question Number 9847 by 0942679167 last updated on 07/Jan/17
Commented by prakash jain last updated on 08/Jan/17
Can u please type the question  image is not readble.  Or use app like camscanner to  take the image
$$\mathrm{Can}\:\mathrm{u}\:\mathrm{please}\:\mathrm{type}\:\mathrm{the}\:\mathrm{question} \\ $$$$\mathrm{image}\:\mathrm{is}\:\mathrm{not}\:\mathrm{readble}. \\ $$$$\mathrm{Or}\:\mathrm{use}\:\mathrm{app}\:\mathrm{like}\:\mathrm{camscanner}\:\mathrm{to} \\ $$$$\mathrm{take}\:\mathrm{the}\:\mathrm{image} \\ $$
Commented by FilupSmith last updated on 09/Jan/17
For n≥2 where n∈N, determine the  largest possible value of the expression:  V_n =sin(x_1 )cos(x_2 )+sin(x_2 )cos(x_3 )+...+sin(x_n )cos(x_1 )  ∀n∈N^(≥2) :x_n ∈R
$$\mathrm{For}\:{n}\geqslant\mathrm{2}\:\mathrm{where}\:{n}\in\mathbb{N},\:\mathrm{determine}\:\mathrm{the} \\ $$$$\mathrm{largest}\:\mathrm{possible}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}: \\ $$$${V}_{{n}} =\mathrm{sin}\left({x}_{\mathrm{1}} \right)\mathrm{cos}\left({x}_{\mathrm{2}} \right)+\mathrm{sin}\left({x}_{\mathrm{2}} \right)\mathrm{cos}\left({x}_{\mathrm{3}} \right)+…+\mathrm{sin}\left({x}_{{n}} \right)\mathrm{cos}\left({x}_{\mathrm{1}} \right) \\ $$$$\forall{n}\in\mathbb{N}^{\geqslant\mathrm{2}} :{x}_{{n}} \in\mathbb{R} \\ $$
Commented by FilupSmith last updated on 09/Jan/17
I believe:  V_n =sin(x_n )cos(x_1 )+Σ_(t=1) ^n sin(x_t )cos(x_(t+1) )
$$\mathrm{I}\:\mathrm{believe}: \\ $$$${V}_{{n}} =\mathrm{sin}\left({x}_{{n}} \right)\mathrm{cos}\left({x}_{\mathrm{1}} \right)+\underset{{t}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{sin}\left({x}_{{t}} \right)\mathrm{cos}\left({x}_{{t}+\mathrm{1}} \right) \\ $$
Answered by mrW1 last updated on 19/Jan/17
max. =n×((√2)/2)×((√2)/2)=(n/2)
$${max}.\:={n}×\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}×\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}=\frac{{n}}{\mathrm{2}} \\ $$

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