Question Number 5081 by FilupSmith last updated on 10/Apr/16 $$\mathrm{Can}\:\mathrm{we}\:\mathrm{prove}: \\ $$$${a}+{a}^{{a}} +{a}^{{a}^{{a}} } +…=\infty,\:\:{a}>\mathrm{1} \\ $$ Answered by Yozzii last updated on 11/Apr/16 $${Define}\:{the}\:{function}\:{f}\:{that}\:{maps}\:…
Question Number 70503 by peter frank last updated on 04/Oct/19 Commented by Prithwish sen last updated on 06/Oct/19 $$\mathrm{Let}\:\mathrm{O}=\left(\mathrm{x}_{\mathrm{1}} ,\mathrm{y}_{\mathrm{1}} \right) \\ $$$$\because\:\mathrm{It}\:\mathrm{is}\:\mathrm{on}\:\mathrm{the}\:\mathrm{circle}\: \\ $$$$\therefore\boldsymbol{\mathrm{x}}_{\mathrm{1}}…
Question Number 136043 by JulioCesar last updated on 18/Mar/21 Answered by rs4089 last updated on 18/Mar/21 $$\int{sec}^{\mathrm{8}} {x}.{dx} \\ $$$$\int\left(\mathrm{1}+{tan}^{\mathrm{2}} {x}\right)^{\mathrm{3}} {sec}^{\mathrm{2}} {x}.{dx} \\ $$$${tanx}={t}\:\Rightarrow{sec}^{\mathrm{2}}…
Question Number 4942 by prakash jain last updated on 24/Mar/16 $$\mathrm{If}\:{p}\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{5},\:\mathrm{the}\:\mathrm{prove} \\ $$$$\mathrm{that}\:{p}\:\mathrm{mod}\:\mathrm{6}\:\equiv\mathrm{1}\:\mathrm{or}\:{p}\:\mathrm{mod}\:\mathrm{6}\equiv\mathrm{5}. \\ $$$$\mathrm{i}.\mathrm{e}.\:\mathrm{All}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{greater}\:\mathrm{than}\:\mathrm{5}\:\mathrm{leave}\:\mathrm{a} \\ $$$$\mathrm{remainder}\:\mathrm{of}\:\mathrm{1}\:\mathrm{or}\:\mathrm{5}\:\mathrm{when}\:\mathrm{divided}\:\mathrm{by}\:\mathrm{6}. \\ $$ Commented by Yozzii last updated on…
Question Number 136005 by Raxreedoroid last updated on 17/Mar/21 $$\mathrm{challenge}\:\mathrm{question} \\ $$$$ \\ $$$$\mathrm{Let}\:{R}\left({m},{n}\right)={R}\left({m},{n}−\mathrm{1}\right)+{R}\left({m}+\mathrm{1},{n}−\mathrm{1}\right) \\ $$$$\mathrm{if}\:{R}\left(\mathrm{1},\mathrm{1}\right)=\mathrm{19},{R}\left(\mathrm{2},\mathrm{1}\right)=\mathrm{9},{R}\left(\mathrm{3},\mathrm{1}\right)=−\mathrm{2} \\ $$$$\mathrm{and}\:\forall{n}\:\mathrm{s}.\mathrm{t}.\:{R}\left(\mathrm{4},{n}\right)=\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$$\left.{a}\right)\:\mathrm{Find}\:{R}\left(\mathrm{1},\mathrm{33}\right) \\…
Question Number 70417 by aliesam last updated on 04/Oct/19 Answered by MJS last updated on 04/Oct/19 $$\mathrm{just}\:\mathrm{use}\:\mathrm{formulas} \\ $$$$\mathrm{sphere}:\:{V}=\frac{\mathrm{4}\pi}{\mathrm{3}}{r}^{\mathrm{3}} ;\:{S}=\mathrm{4}\pi{r}^{\mathrm{2}} \\ $$$$\mathrm{cylinder}:\:{V}=\pi{r}^{\mathrm{2}} {h} \\ $$$$\:\:\:\:\:{S}=\mathrm{2}×\mathrm{circle}+\mathrm{lateral}=\mathrm{2}\pi{r}^{\mathrm{2}}…
Question Number 70399 by Henri Boucatchou last updated on 04/Oct/19 $$\boldsymbol{{Solve}}\:\:\boldsymbol{{x}}^{\mathrm{4}} \:+\:\boldsymbol{{x}}^{\mathrm{3}} \:−\mathrm{2}\boldsymbol{{ax}}^{\mathrm{2}} \:−\boldsymbol{{ax}}\:+\:\boldsymbol{{a}}^{\mathrm{2}} =\:\mathrm{0},\:\:\boldsymbol{{a}}\:\in\:\mathbb{R} \\ $$ Commented by Prithwish sen last updated on 04/Oct/19…
Question Number 135889 by Ar Brandon last updated on 16/Mar/21 $$\mathrm{1}\backslash\:\mathrm{If}\:\mathrm{A}\:\mathrm{and}\:\mathrm{B}\:\mathrm{are}\:\mathrm{sets}\:\mathrm{define}\:\mathrm{their}\:\mathrm{scheffer}\:\mathrm{product}\:\mathrm{A}\ast\mathrm{B}\:\mathrm{by}\:\mathrm{A}\ast\mathrm{B}=\mathrm{A}\ast\cap\mathrm{B}\ast \\ $$$$\mathrm{Prove}\:\mathrm{by}\:\mathrm{definitions}\:\mathrm{that}\:\left(\mathrm{A}\ast\mathrm{B}\right)\ast\left(\mathrm{A}\ast\mathrm{B}\right)=\mathrm{A}\cup\mathrm{B} \\ $$$$ \\ $$$$\mathrm{2}\backslash\:\mathrm{State}\:\mathrm{the}\:\mathrm{strong}\:\mathrm{principle}\:\mathrm{of}\:\mathrm{mathematical}\:\mathrm{induction}. \\ $$$$\mathrm{Suppose}\:\mathrm{that}\:\mathrm{a}_{\mathrm{1}} =\mathrm{1}\:,\:\mathrm{a}_{\mathrm{2}} =\mathrm{3} \\ $$$$\mathrm{a}_{\mathrm{k}} =\mathrm{a}_{\mathrm{k}−\mathrm{2}} +\mathrm{2a}_{\mathrm{k}−\mathrm{1}}…
Question Number 4785 by Dnilka228 last updated on 10/Mar/16 $${n}^{{i}} \frac{{n}}{{i}×{i}}\approx\mathrm{sin}\:{n}^{{i}} +\alpha \\ $$ Commented by Dnilka228 last updated on 10/Mar/16 $${n}^{{i}} \frac{{n}}{{i}×{i}}\neq\mathrm{sin}\:\alpha+{n}^{{i}} \\ $$$$\sqrt{{n}^{{i}}…
Question Number 4704 by lakshaysethi039 last updated on 25/Feb/16 $${If}\:{a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ,………{a}_{{n}} {be}\:{an}\:{arithmetic}\:{progression}, \\ $$$${then}\:{show}\:{that} \\ $$$$\frac{\mathrm{1}}{{a}_{\mathrm{1}} {a}_{{n}} }\:+\:\frac{\mathrm{1}}{{a}_{\mathrm{2}} {a}_{{n}−\mathrm{1}} }\:+\:\frac{\mathrm{1}}{{a}_{\mathrm{3}} {a}_{{n}−\mathrm{2}} }\:+………….+\frac{\mathrm{1}}{{a}_{{n}} {a}_{\mathrm{1}} }\:…