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Question-212075

Question Number 212075 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\left(\sqrt{{n}+\sqrt{{n}}}−\sqrt{{n}}\right)\:=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+\sqrt{{t}}}−\mathrm{1}}{\:\sqrt{{t}}}\:= \\ $$$$\:\:\:\:\:\left[\mathrm{l}'\mathrm{H}\hat {\mathrm{o}pital}\right] \\ $$$$=\underset{{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{2}\sqrt{\mathrm{1}+\sqrt{{t}}}}\:=\frac{\mathrm{1}}{\mathrm{2}}…

HCF-of-n-2-10-n-1-2-10-n-N-

Question Number 212043 by RojaTaniya last updated on 28/Sep/24 $$\:{HCF}\:{of}\:\left\{\left({n}^{\mathrm{2}} +\mathrm{10}\right),\:\left({n}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{10}\right\}=? \\ $$$$\:\:{n}\in{N} \\ $$ Answered by A5T last updated on 28/Sep/24 $${Let}\:{a}={n}^{\mathrm{2}} +\mathrm{10},\:{b}=\left({n}+\mathrm{1}\right)^{\mathrm{2}}…

Question-212066

Question Number 212066 by Spillover last updated on 28/Sep/24 Answered by Rasheed.Sindhi last updated on 29/Sep/24 $$\frac{\mathrm{2021}!+\mathrm{2020}!}{\mathrm{2021}!−\mathrm{2020}!}\:\centerdot\:\frac{\mathrm{2020}!+\mathrm{2019}!}{\mathrm{2020}!−\mathrm{2019}!}\:\centerdot…\frac{\mathrm{3}!+\mathrm{2}!}{\mathrm{3}!−\mathrm{2}!}\:\centerdot\:\frac{\mathrm{2}!+\mathrm{1}!}{\mathrm{2}!−\mathrm{1}!} \\ $$$$\frac{\left\{\left({x}+\mathrm{1}\right)!+{x}!\right\}}{\left\{\left({x}+\mathrm{1}\right)!−{x}!\right\}}=\frac{{x}!\left\{\left({x}+\mathrm{1}\right)+\mathrm{1}\right\}}{{x}!\left\{\left({x}+\mathrm{1}\right)−\mathrm{1}\right\}}=\frac{{x}+\mathrm{2}}{{x}} \\ $$$$\frac{\mathrm{2020}+\mathrm{2}}{\mathrm{2020}}\centerdot\frac{\mathrm{2019}+\mathrm{2}}{\mathrm{2019}}\centerdot…\frac{\mathrm{2}+\mathrm{2}}{\mathrm{2}}\centerdot\frac{\mathrm{1}+\mathrm{2}}{\mathrm{1}} \\ $$$$\frac{\left(\mathrm{2020}+\mathrm{2}\right)\left(\mathrm{2019}+\mathrm{2}\right)\left(\mathrm{2018}+\mathrm{2}\right)…\left(\mathrm{2}+\mathrm{2}\right)\left(\mathrm{1}+\mathrm{2}\right)}{\mathrm{2020}.\mathrm{2019}.\mathrm{2018}….\mathrm{2}.\mathrm{1}} \\ $$$$=\frac{\mathrm{2022}.\mathrm{2021}.\mathrm{2020}….\mathrm{4}.\mathrm{3}}{\mathrm{2020}!}…

DEFINATION-OF-QUADRATIC-FORM-A-Quadratic-form-is-a-homogeneous-polynomial-of-degree-two-in-multiple-variable-Q-X-T-AX-Here-Q-Quadratic-form-ax-2-by-2-cz

Question Number 212028 by siva12345 last updated on 27/Sep/24 $${DEFINATION}\:\:\:\:{OF}\:\:\:{QUADRATIC}\:\:{FORM}:\: \\ $$$$\:\:\:\:\:{A}\:\:{Quadratic}\:\:{form}\:\:{is}\:\:{a}\:{homogeneous}\:\:{polynomial}\:\:{of}\:\:{degree}\:{two}\:\:{in}\:\:{multiple}\:\:{variable}.\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Q}={X}^{{T}} {AX} \\ $$$${Here}\:\:{Q}={Quadratic}\:{form}. \\ $$$${ax}^{\mathrm{2}} +{by}^{\mathrm{2}} +{cz}^{\mathrm{2}} +\mathrm{2}{hxy}+\mathrm{2}{fyz}+\mathrm{2}{gzx}=\mathrm{0} \\ $$$${By}\:\:{using}\:\:{these}\:\:{Q}={X}^{{T}} {AX}\:\:\left[{we}\:\:{can}\:\:{write}\:{matrix}\:{A}\right]…