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for-all-positive-integral-u-n-1-u-n-u-n-1-2-2-u-n-u-n-2-and-u-1-2-1-2-prove-that-3log-2-u-n-2-n-1-1-n-where-x-is-the-integral-part-of-x-




Question Number 143474 by Ghaniy last updated on 14/Jun/21
for all positive integral.,   u_(n+1) =u_n (u_(n−1) ^2 −2)−u_n    u_n =2 and u_1 =2(1/2)  prove that : 3log_2 [u_n ]=2^n −1(−1)^n   where [x] is the integral part of  x
$${for}\:{all}\:{positive}\:{integral}., \\ $$$$\:\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{u}_{\mathrm{n}} \left(\mathrm{u}_{\mathrm{n}−\mathrm{1}} ^{\mathrm{2}} −\mathrm{2}\right)−\mathrm{u}_{\mathrm{n}} \\ $$$$\:\mathrm{u}_{\mathrm{n}} =\mathrm{2}\:{and}\:\mathrm{u}_{\mathrm{1}} =\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${prove}\:{that}\::\:\mathrm{3log}_{\mathrm{2}} \left[\mathrm{u}_{\mathrm{n}} \right]=\mathrm{2}^{\mathrm{n}} −\mathrm{1}\left(−\mathrm{1}\right)^{\mathrm{n}} \\ $$$${where}\:\left[\mathrm{x}\right]\:{is}\:{the}\:{integral}\:{part}\:{of}\:\:\mathrm{x} \\ $$

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