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Question Number 49640 by maxmathsup by imad last updated on 08/Dec/18
if x ∈[p,(√(p^2  +2))]  calculate [x]
$${if}\:{x}\:\in\left[{p},\sqrt{{p}^{\mathrm{2}} \:+\mathrm{2}}\right]\:\:{calculate}\:\left[{x}\right] \\ $$
Commented by maxmathsup by imad last updated on 08/Dec/18
p from N
$${p}\:{from}\:{N} \\ $$
Commented by Abdo msup. last updated on 23/Dec/18
we have p≤x≤(√(p^2  +2))≤p+1 ⇒[x]=p
$${we}\:{have}\:{p}\leqslant{x}\leqslant\sqrt{{p}^{\mathrm{2}} \:+\mathrm{2}}\leqslant{p}+\mathrm{1}\:\Rightarrow\left[{x}\right]={p} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 09/Dec/18
consider p=+ve integer  p=I  (√(p^2 +2)) =I+f  when x lies between [p,(√(p^2 +2)) ]                                              =[I,I+f]  [x]=I  that implies [x]=p
$${consider}\:{p}=+{ve}\:{integer} \\ $$$${p}={I} \\ $$$$\sqrt{{p}^{\mathrm{2}} +\mathrm{2}}\:={I}+{f} \\ $$$${when}\:{x}\:{lies}\:{between}\:\left[{p},\sqrt{{p}^{\mathrm{2}} +\mathrm{2}}\:\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left[{I},{I}+{f}\right] \\ $$$$\left[{x}\right]={I}\:\:{that}\:{implies}\:\left[{x}\right]={p} \\ $$

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