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](Q143474.png)
$${for}\:{all}\:{positive}\:{integral}., \\ $$$$\:\mathrm{u}_{\mathrm{n}+\mathrm{1}} =\mathrm{u}_{\mathrm{n}} \left(\mathrm{u}_{\mathrm{n}−\mathrm{1}} ^{\mathrm{2}} −\mathrm{2}\right)−\mathrm{u}_{\mathrm{n}} \\ $$$$\:\mathrm{u}_{\mathrm{n}} =\mathrm{2}\:{and}\:\mathrm{u}_{\mathrm{1}} =\mathrm{2}\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${prove}\:{that}\::\:\mathrm{3log}_{\mathrm{2}} \left[\mathrm{u}_{\mathrm{n}} \right]=\mathrm{2}^{\mathrm{n}} −\mathrm{1}\left(−\mathrm{1}\right)^{\mathrm{n}} \\ $$$${where}\:\left[\mathrm{x}\right]\:{is}\:{the}\:{integral}\:{part}\:{of}\:\:\mathrm{x} \\ $$