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Question Number 157660 by naka3546 last updated on 26/Oct/21

Given  x_1  = 1, x_2  , x_3  , …, is  a  real  numbers  sequence  for  n ≥ 1  with    recurrence  relation  x_(n+1)  − x_n  = (1/(2x_n ))  .  [x]  is  expressed  as  the  largest  integer  of  x  .  [25x_(625) ]  =  ?

$${Given}\:\:{x}_{\mathrm{1}} \:=\:\mathrm{1},\:{x}_{\mathrm{2}} \:,\:{x}_{\mathrm{3}} \:,\:\ldots,\:{is}\:\:{a}\:\:{real}\:\:{numbers}\:\:{sequence}\:\:{for}\:\:{n}\:\geqslant\:\mathrm{1}\:\:{with}\:\: \\ $$$${recurrence}\:\:{relation}\:\:{x}_{{n}+\mathrm{1}} \:−\:{x}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{2}{x}_{{n}} }\:\:. \\ $$$$\left[{x}\right]\:\:{is}\:\:{expressed}\:\:{as}\:\:{the}\:\:{largest}\:\:{integer}\:\:{of}\:\:{x}\:\:. \\ $$$$\left[\mathrm{25}{x}_{\mathrm{625}} \right]\:\:=\:\:? \\ $$

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