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knowns-x-2-2-x-1-x-3-3-x-2-x-4-4-x-3-x-5-5-x-4-x-8-8-x-7-Find-the-value-of-x-1-x-2-x-3-x-8-




Question Number 152052 by john_santu last updated on 25/Aug/21
  knowns x_2 =(2/x_1 ) , x_3 =(3/x_2 ) , x_4 =(4/x_3 )  , x_5 =(5/x_4 ) , ..., x_8 =(8/x_7 ). Find the  value of x_1 ×x_2 ×x_3 ×...×x_8 .
$$\:\:\mathrm{knowns}\:\mathrm{x}_{\mathrm{2}} =\frac{\mathrm{2}}{\mathrm{x}_{\mathrm{1}} }\:,\:\mathrm{x}_{\mathrm{3}} =\frac{\mathrm{3}}{\mathrm{x}_{\mathrm{2}} }\:,\:\mathrm{x}_{\mathrm{4}} =\frac{\mathrm{4}}{\mathrm{x}_{\mathrm{3}} } \\ $$$$,\:\mathrm{x}_{\mathrm{5}} =\frac{\mathrm{5}}{\mathrm{x}_{\mathrm{4}} }\:,\:…,\:\mathrm{x}_{\mathrm{8}} =\frac{\mathrm{8}}{\mathrm{x}_{\mathrm{7}} }.\:\mathrm{Find}\:\mathrm{the} \\ $$$$\mathrm{value}\:\mathrm{of}\:\mathrm{x}_{\mathrm{1}} ×\mathrm{x}_{\mathrm{2}} ×\mathrm{x}_{\mathrm{3}} ×…×\mathrm{x}_{\mathrm{8}} . \\ $$
Answered by Olaf_Thorendsen last updated on 25/Aug/21
P = x_1 x_2 x_3 x_4 x_5 x_6 x_7 x_8   P = x_1 (2/x_1 )x_3 (4/x_3 )x_5 (6/x_5 )x_7 (8/x_7 )  P = 2×4×6×8 = 384
$$\mathrm{P}\:=\:{x}_{\mathrm{1}} {x}_{\mathrm{2}} {x}_{\mathrm{3}} {x}_{\mathrm{4}} {x}_{\mathrm{5}} {x}_{\mathrm{6}} {x}_{\mathrm{7}} {x}_{\mathrm{8}} \\ $$$$\mathrm{P}\:=\:{x}_{\mathrm{1}} \frac{\mathrm{2}}{{x}_{\mathrm{1}} }{x}_{\mathrm{3}} \frac{\mathrm{4}}{{x}_{\mathrm{3}} }{x}_{\mathrm{5}} \frac{\mathrm{6}}{{x}_{\mathrm{5}} }{x}_{\mathrm{7}} \frac{\mathrm{8}}{{x}_{\mathrm{7}} } \\ $$$$\mathrm{P}\:=\:\mathrm{2}×\mathrm{4}×\mathrm{6}×\mathrm{8}\:=\:\mathrm{384} \\ $$

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