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Category: Algebra

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Question Number 194634 by York12 last updated on 12/Jul/23 $${a}_{\mathrm{1}} ,{a}_{\mathrm{2}} ,{a}_{\mathrm{3}} ,….,{a}_{{n}} >\mathrm{0}\:{such}\:{that}\:{a}_{{i}} \in\left[\mathrm{0},{i}\right]\: \\ $$$$\forall\:{i}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},…,{n}\right\}\:{prove}\:{that} \\ $$$$\mathrm{2}^{{n}} .{a}_{\mathrm{1}} \left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} \right)…\left({a}_{\mathrm{1}} +{a}_{\mathrm{2}} +…+{a}_{{n}}…

Question-194612

Question Number 194612 by Abdullahrussell last updated on 11/Jul/23 Commented by TheHoneyCat last updated on 15/Jul/23 $$\left.\mathrm{1}\right)\:\mathrm{Let}\:\alpha=\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+\mathrm{8}+\mathrm{9}=\mathrm{45} \\ $$$$\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{basis}\:\mathrm{10}. \\ $$$$\mathrm{Let}\:{S}_{\mathrm{1}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{first}\:\mathrm{sum},\:{S}_{\mathrm{1},\mathrm{0}} \:\mathrm{be}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{all} \\ $$$$\mathrm{unit}\:\mathrm{digits},\:{S}_{\mathrm{1},\mathrm{1}}…

where-can-I-learn-about-multiple-sigma-notaions-of-dependent-and-independent-variables-something-like-this-1-i-lt-j-lt-k-1-i-j-k-find-I-want-to-know-what-to-study-

Question Number 194610 by justenspi last updated on 11/Jul/23 $${where}\:{can}\:{I}\:{learn}\:{about}\:{multiple}\:{sigma}\:{notaions} \\ $$$${of}\:{dependent}\:{and}\:{independent}\:{variables} \\ $$$$ \\ $$$${something}\:{like}\:{this} \\ $$$$\underset{\mathrm{1}\leqslant{i}} {\sum}\underset{<{j}} {\sum}\underset{<{k}\leqslant\mathrm{1}} {\sum}\left({i}+{j}+{k}\right)=\lambda \\ $$$${find}\:\lambda \\ $$$${I}\:{want}\:{to}\:{know}\:{what}\:{to}\:{study}…

if-u-n-1-5-1-5-2-n-1-5-2-n-then-u-n-1-u-n-u-n-1-n-0-1-2-

Question Number 194579 by MM42 last updated on 10/Jul/23 $${if}\:\:\:\:{u}_{{n}} =\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}}}\left[\left(\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}} −\left(\frac{\mathrm{1}−\sqrt{\mathrm{5}}}{\mathrm{2}}\right)^{{n}} \right] \\ $$$$\:{then}\:\:\:{u}_{{n}+\mathrm{1}} ={u}_{{n}} +{u}_{{n}−\mathrm{1}} \:\:\:?\:\:\:\:\:;\:\:\:{n}=\mathrm{0},\mathrm{1},\mathrm{2},.. \\ $$ Answered by Frix last updated…

f-x-1-f-x-x-2-f-x-f-6-f-3-

Question Number 194526 by mathlove last updated on 09/Jul/23 $$\frac{{f}\left({x}+\mathrm{1}\right)}{{f}\left({x}\right)}={x}^{\mathrm{2}\:\:\:\:\:\:\:\:} \:\:\:\:{f}\left({x}\right)=? \\ $$$$\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{3}\right)}=? \\ $$ Answered by JDamian last updated on 09/Jul/23 $$\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{3}\right)}=\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{3}\right)}×\frac{{f}\left(\mathrm{5}\right)}{{f}\left(\mathrm{5}\right)}×\frac{{f}\left(\mathrm{4}\right)}{{f}\left(\mathrm{4}\right)}= \\ $$$$=\frac{{f}\left(\mathrm{6}\right)}{{f}\left(\mathrm{5}\right)}×\frac{{f}\left(\mathrm{5}\right)}{{f}\left(\mathrm{4}\right)}×\frac{{f}\left(\mathrm{4}\right)}{{f}\left(\mathrm{3}\right)}=\mathrm{5}^{\mathrm{2}}…