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Question Number 6340 by sanusihammed last updated on 24/Jun/16 Commented by nburiburu last updated on 24/Jun/16 $${seems}\:{it}\:{need}\:{more}\:{info}.\:{For}\:{the}\:{moment}\:{being}\:{could}\:{be}\:{that}\:{moving}\:{one}\:{unit}\:{from}\:{right}\:{to}\:{left}\:{adds}\:\mathrm{2}\:{units}\:{up}\:{and}\:{cost}\:\mathrm{1}\:{in}\:{the}\:{centre}. \\ $$$${However}\:{many}\:{other}\:{posibilities}\:{could}\:{explain}\:{the}\:{numbers}\:{in}\:{the}\:{diagram}. \\ $$ Commented by prakash jain…
Question Number 6311 by FilupSmith last updated on 23/Jun/16 $$\mathrm{According}\:\mathrm{to}\:\mathrm{WolframAlpha}: \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\prod}}\left(\mathrm{1}−{x}^{\left(−\mathrm{1}\right)^{{k}} } \right)=\left(\mathrm{1}−{x}\right)^{\lfloor\frac{{n}}{\mathrm{2}}\rfloor+\mathrm{1}} \left(\frac{{x}−\mathrm{1}}{{x}}\right)^{\lfloor\frac{{n}−\mathrm{1}}{\mathrm{2}}\rfloor+\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{Can}\:\mathrm{anyone}\:\mathrm{work}\:\mathrm{out}\:\mathrm{how}? \\ $$ Commented by…
Question Number 6289 by sanusihammed last updated on 22/Jun/16 Commented by FilupSmith last updated on 22/Jun/16 $${a}=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{5}}{\mathrm{6}}+…+\frac{\mathrm{2015}}{\mathrm{2016}} \\ $$$${b}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{4}}{\mathrm{6}}×…×\frac{\mathrm{2015}}{\mathrm{2016}} \\ $$$$\frac{{a}}{{b}}+\frac{{b}}{{a}}=?? \\ $$$$ \\ $$$${a}=\underset{{n}=\mathrm{1}}…
Question Number 71790 by jatin123 last updated on 20/Oct/19 Answered by $@ty@m123 last updated on 20/Oct/19 $$=\frac{\mathrm{2}}{\mathrm{3}}×\frac{\mathrm{3}}{\mathrm{4}}×…..×\frac{\mathrm{98}}{\mathrm{99}}×\frac{\mathrm{99}}{\mathrm{100}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{100}}=\frac{\mathrm{1}}{\mathrm{50}} \\ $$ Commented by jatin123 last…
Question Number 137324 by greg_ed last updated on 01/Apr/21 $$\boldsymbol{\mathrm{how}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{evaluate}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{one}}\:: \\ $$$$\mathrm{P}\:=\:\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1958}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1959}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{1960}}\right)…\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2017}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2018}}\right)\left(\mathrm{1}+\:\frac{\mathrm{1}}{\mathrm{2019}}\right) \\ $$$$\boldsymbol{\mathrm{P}}\:=\:?\: \\ $$ Answered by som(math1967) last updated on 01/Apr/21 $${P}=\left(\frac{\mathrm{1959}}{\mathrm{1958}}\right)\left(\frac{\mathrm{1960}}{\mathrm{1959}}\right)\left(\frac{\mathrm{1961}}{\mathrm{1960}}\right)..\left(\frac{\mathrm{2019}}{\mathrm{2018}}\right)\left(\frac{\mathrm{2020}}{\mathrm{2019}}\right) \\…
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Question Number 71717 by peter frank last updated on 19/Oct/19 Commented by mathmax by abdo last updated on 19/Oct/19 $${let}\:{I}=\int\:\frac{{dx}}{{cos}\left({x}−{a}\right){cos}\left({x}−{b}\right)}\:\Rightarrow{I}=\int\:\:\frac{\mathrm{2}{dx}}{{cos}\left(\mathrm{2}{x}−{a}−{b}\right)+{cos}\left({a}−{b}\right)} \\ $$$$=_{\mathrm{2}{x}−\left({a}+{b}\right)={t}} \:\:\:\:\int\:\:\frac{{dt}}{{cost}\:+\lambda}\:\:{with}\:\lambda={cos}\left({a}−{b}\right)\:{changement} \\ $$$${tan}\left(\frac{{t}}{\mathrm{2}}\right)={u}\:{give}\:\int\:\:\:\frac{{dt}}{{cost}\:+\lambda}\:=\int\:\:\frac{\mathrm{1}}{\frac{\mathrm{1}−{u}^{\mathrm{2}}…
Question Number 6166 by sanusihammed last updated on 16/Jun/16 $${If}\:{the}\:{sum}\:{of}\:{the}\:{first}\:{nth}\:{terms}\:{of}\:{a}\:{sequence}\:{is}\:{given}\:{by}\: \\ $$$${S}_{{n}\:} \:=\:\:\mathrm{9}\left(\mathrm{1}\:−\:\frac{\mathrm{1}}{\mathrm{3}^{{n}} \:}\right) \\ $$$$\left({a}\right)\:{find}\:{the}\:{first}\:{and}\:{the}\:{second}\:{term}\:{of}\:{the}\:{sequence} \\ $$$$\left({b}\right)\:{find}\:{the}\:{nth}\:{term}\:{of}\:{the}\:{sequence} \\ $$$$\left({c}\right)\:{show}\:{that}\:{the}\:{sequence}\:{is}\:{a}\:{GP}\:{and}\:{find}\:{it}\:{common}\:{ratio} \\ $$$$ \\ $$$${please}\:{help}. \\…
Question Number 6158 by enigmeyou last updated on 16/Jun/16 $$\frac{−\mathrm{1}}{\mathrm{1048576}}=\frac{\mathrm{1}}{\mathrm{8}}×\left(\frac{−\mathrm{1}}{\mathrm{2}}\right)^{{n}} \\ $$$${find}\:{n}\:? \\ $$ Answered by Rasheed Soomro last updated on 16/Jun/16 $$\frac{−\mathrm{1}}{\mathrm{1048576}}=\frac{\mathrm{1}}{\mathrm{8}}×\left(\frac{−\mathrm{1}}{\mathrm{2}}\right)^{{n}} \\ $$$$\frac{−\mathrm{8}}{\mathrm{1048576}}=\left(\frac{−\mathrm{1}}{\mathrm{2}}\right)^{{n}}…