Question Number 213802 by efronzo1 last updated on 17/Nov/24 Answered by A5T last updated on 17/Nov/24 $${t}_{\mathrm{1}} ={k}−\mathrm{1};{t}_{\mathrm{2}} ={k} \\ $$$${t}_{\mathrm{3}} =\frac{\mathrm{5}{k}+\mathrm{1}}{\mathrm{25}{k}−\mathrm{25}};\:\:\:\:{t}_{\mathrm{4}} =\frac{\frac{\mathrm{10}{k}−\mathrm{4}}{\mathrm{5}{k}−\mathrm{5}}}{\mathrm{25}{k}}=\frac{\mathrm{10}{k}−\mathrm{4}}{\mathrm{125}{k}\left({k}−\mathrm{1}\right)} \\ $$$${t}_{\mathrm{5}}…
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Question Number 213522 by efronzo1 last updated on 07/Nov/24 $$\:\:\mathrm{The}\:\mathrm{two}\:\mathrm{corner}\:\mathrm{points}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\: \\ $$$$\:\:\mathrm{lie}\:\mathrm{on}\:\mathrm{curve}\:\mathrm{f}\left(\mathrm{x}\right)=\:\mathrm{x}^{\mathrm{2}} −\mathrm{2x}−\mathrm{3}\:\mathrm{and}\: \\ $$$$\:\mathrm{the}\:\mathrm{other}\:\mathrm{two}\:\mathrm{corner}\:\mathrm{points}\:\mathrm{lie}\:\mathrm{on}\: \\ $$$$\:\mathrm{curve}\:\mathrm{g}\left(\mathrm{x}\right)=\:−\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{3}\:.\:\mathrm{It}\:\mathrm{is}\:\mathrm{known} \\ $$$$\:\mathrm{that}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{a}\:\mathrm{square}\:\mathrm{can}\:\mathrm{be}\: \\ $$$$\:\:\mathrm{expressed}\:\mathrm{by}\:\mathrm{p}+\mathrm{q}\sqrt{\mathrm{r}}\:,\:\mathrm{for}\:\mathrm{a}\:\mathrm{natural}\: \\ $$$$\:\mathrm{number}\:\mathrm{p},\mathrm{q}\:,\mathrm{r}\:\mathrm{where}\:\mathrm{r}\:\mathrm{is}\:\mathrm{not}\:\mathrm{divisible}\: \\…
Question Number 213261 by lmcp1203 last updated on 02/Nov/24 $${if}\:{between}\:\:\mathrm{10}^{\mathrm{4}} \:{and}\:\mathrm{10}^{{n}} \:{there}\:{are}\:\mathrm{9999000}\:{coprime}\:{numbers}\:{with}\:\mathrm{20}.\:{find}\:{n}.\:\:{please}.\:{thanks} \\ $$ Answered by MrGaster last updated on 02/Nov/24 $$\mathrm{10}^{\mathrm{4}} <{x}<\mathrm{10}^{{n}} ,\left({x},\mathrm{20}\right)=\mathrm{1} \\…
Question Number 213100 by MathematicalUser2357 last updated on 30/Oct/24 $$\mathrm{Find}\:\mathrm{this}\:\mathrm{numeric}\:\mathrm{expression}\:\mathrm{using}: \\ $$$$\mathrm{The}\:\mathrm{arithmetic}\:\mathrm{division}\:\mathrm{rule}\:{a}\boldsymbol{\div}{b}\left({c}\right)={a}\boldsymbol{\div}{b}×{c}, \\ $$$$\mathrm{The}\:\mathrm{solvable}\:\mathrm{incorrect}\:\mathrm{syntax}\:\mathrm{rule}\:\left({a}\right){b}={a}×{b},\:\mathrm{where}\:{b}\:\mathrm{is}\:\mathrm{a}\:\mathrm{number}: \\ $$$$\mathrm{12}\boldsymbol{\div}\mathrm{4}\left(\mathrm{10}−\mathrm{8}+\mathrm{1}\right)\mathrm{2}\boldsymbol{\div}\mathrm{6}×\mathrm{2}=? \\ $$ Commented by A5T last updated on 30/Oct/24…
Question Number 213000 by golsendro last updated on 28/Oct/24 $$\:\:\:\mathrm{f}\left(\frac{\mathrm{x}−\mathrm{3}}{\mathrm{x}+\mathrm{1}}\right)\:+\:\mathrm{f}\left(\frac{\mathrm{x}+\mathrm{3}}{\mathrm{1}−\mathrm{x}}\right)\:=\:\mathrm{x}\:,\:\mathrm{x}\neq\:\pm\:\mathrm{1} \\ $$$$\:\:\:\mathrm{f}\left(\mathrm{x}\right)=? \\ $$ Answered by Ghisom last updated on 28/Oct/24 $$\frac{{u}−\mathrm{3}}{{u}+\mathrm{1}}={v}\:\Leftrightarrow\:{u}=\frac{{v}+\mathrm{3}}{\mathrm{1}−{v}} \\ $$$$\mathrm{2}{f}\left({x}\right)=\frac{{x}−\mathrm{3}}{{x}+\mathrm{1}}+\frac{{x}+\mathrm{3}}{\mathrm{1}−{x}}−{x} \\…
Question Number 212496 by efronzo1 last updated on 15/Oct/24 $$\:\mathrm{If}\:\mathrm{1}.\mathrm{1}!+\mathrm{3}.\mathrm{2}!+…+\left(\mathrm{2n}−\mathrm{1}\right).\mathrm{n}!\: \\ $$$$\:=\:\mathrm{a}.\left(\mathrm{n}+\mathrm{1}\right)!+\mathrm{b}\left(\mathrm{1}!+\mathrm{2}!+…+\left(\mathrm{n}+\mathrm{1}\right)!\right)+\mathrm{c}\: \\ $$$$\:\mathrm{for}\:\mathrm{a},\mathrm{b},\mathrm{c}\:\mathrm{integers}\:\mathrm{number} \\ $$$$\:\mathrm{find}\:\mathrm{2a}−\mathrm{b}+\mathrm{3c}\: \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 212050 by Spillover last updated on 28/Sep/24 Answered by Ghisom last updated on 28/Sep/24 $$\int\frac{{x}}{\:\sqrt{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right)}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\mathrm{2}\sqrt{\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\right] \\ $$$$=\int{dt}={t}= \\…
Question Number 211979 by Spillover last updated on 25/Sep/24 Answered by BHOOPENDRA last updated on 25/Sep/24 $$\int\frac{{dx}}{\left({x}^{\mathrm{2}} \mathrm{tan}^{−\mathrm{1}} {x}+\mathrm{tan}^{−\mathrm{1}} {x}\:+{x}^{\mathrm{2}} \pi+\pi\right)} \\ $$$$\int\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left(\mathrm{tan}^{−\mathrm{1}} {x}+\pi\right)}…
Question Number 211920 by Spillover last updated on 24/Sep/24 Answered by aleks041103 last updated on 24/Sep/24 $$\sqrt{{x}\sqrt{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} \sqrt{…}}}\:}=\:{x}^{\mathrm{1}/\mathrm{2}} {x}^{\mathrm{2}/\mathrm{4}} {x}^{\mathrm{3}/\mathrm{8}} …{x}^{{n}/\mathrm{2}^{{n}} } …\:= \\…