Question Number 198304 by universe last updated on 17/Oct/23 $$\:\:\:\mathrm{for}\:\left\{\mathrm{a}_{\mathrm{n}} \right\}\:\mathrm{be}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{of}\:\mathrm{positive}\:\mathrm{real}\:\mathrm{numbers} \\ $$$$\:\:\:\mathrm{such}\:\mathrm{that}\:\:\mathrm{a}_{\mathrm{1}} =\mathrm{1}\:,\:\mathrm{a}_{\mathrm{n}+\mathrm{1}} ^{\mathrm{2}} −\mathrm{2a}_{\mathrm{n}} \mathrm{a}_{\mathrm{n}+\mathrm{1}} −\mathrm{a}_{\mathrm{n}} \:=\:\mathrm{0}\:,\:\forall\:\mathrm{n}\geqslant\:\mathrm{1} \\ $$$$\:\:\:\mathrm{than}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{series}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{a}_{\mathrm{n}} }{\mathrm{3}^{\mathrm{n}\:} }\:\:\mathrm{lies}\:\mathrm{in}\:\mathrm{the}\:\mathrm{interval}…
Question Number 198293 by Mingma last updated on 16/Oct/23 Answered by MM42 last updated on 17/Oct/23 $${B}^{{n}} =\begin{bmatrix}{\left(−\mathrm{1}\right)^{{n}} \:\:\:\:\mathrm{0}}\\{\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{2}^{{n}} }\end{bmatrix}\:\&\:\:\left({ABA}^{−\mathrm{1}} \right)^{{n}} ={AB}^{{n}} {A}^{−\mathrm{1}} \\ $$$$\Rightarrow\left({ABA}^{−\mathrm{1}}…
Question Number 198266 by Mingma last updated on 15/Oct/23 Answered by mr W last updated on 16/Oct/23 $${a}\left(−\mathrm{4}{x}+\mathrm{3}{y}+\mathrm{4}{z}−\mathrm{3}\right)+{b}\left(−\mathrm{2}{x}+\mathrm{4}{y}+\mathrm{5}{z}−\mathrm{5}\right)=\mathrm{10}{x}−\mathrm{11}{y}+{hz}−{k} \\ $$$$\left(−\mathrm{4}{a}−\mathrm{2}{b}−\mathrm{10}\right){x}+\left(\mathrm{3}{a}+\mathrm{4}{b}+\mathrm{11}\right){y}+\left(\mathrm{4}{a}+\mathrm{5}{b}−{h}\right){z}−\mathrm{3}{a}−\mathrm{5}{b}+{k}=\mathrm{0} \\ $$$$−\mathrm{4}{a}−\mathrm{2}{b}−\mathrm{10}=\mathrm{0} \\ $$$$\mathrm{3}{a}+\mathrm{4}{b}+\mathrm{11}=\mathrm{0} \\…
Question Number 198267 by Fridunatjan08 last updated on 16/Oct/23 $${Find}\:{the}\:{real}\:{values}\:{of}\:{n}:\:{n}^{\mathrm{6}} −{n}^{\mathrm{3}} =\mathrm{2} \\ $$ Answered by Rasheed.Sindhi last updated on 16/Oct/23 $${n}^{\mathrm{6}} −{n}^{\mathrm{3}} =\mathrm{2} \\…
Question Number 198252 by hardmath last updated on 15/Oct/23 Commented by JDamian last updated on 15/Oct/23 what does / / mean? Commented by hardmath last updated on 15/Oct/23 $$\bigtriangleup\mathrm{ABC}…
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Question Number 198243 by mr W last updated on 15/Oct/23 $${find}\:{the}\:{sum}\:{of}\:{the}\:{first}\:{n}\:{terms}\:{from} \\ $$$$\mathrm{1},\:\mathrm{2}+\mathrm{3},\:\mathrm{4}+\mathrm{5}+\mathrm{6},\:\mathrm{7}+\mathrm{8}+\mathrm{9}+\mathrm{10},\:… \\ $$ Answered by som(math1967) last updated on 15/Oct/23 $$\mathrm{1}+\mathrm{2}+\mathrm{3}+\mathrm{4}+\mathrm{5}+\mathrm{6}+\mathrm{7}+\mathrm{8}+\mathrm{9}+\mathrm{10}+..{n} \\ $$$${no}\:{of}\:{term}\:=\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}…
Question Number 198197 by necx122 last updated on 13/Oct/23 $${please}\:{helpe} \\ $$$${sinz}\:=\:\mathrm{2}.\:{Find}\:{z} \\ $$ Commented by mokys last updated on 13/Oct/23 Answered by mr W…
Question Number 198175 by York12 last updated on 12/Oct/23 $${Prove}\:{The}\:{following}\:{Functional}\:{equation}: \\ $$$$\zeta\left({x},{s}\right)=\frac{\mathrm{2}\Gamma\left(\mathrm{1}−{s}\right)}{\left(\mathrm{2}\pi\right)^{\left(\mathrm{1}−{s}\right)} }\left\{{sin}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{cos}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]+{cos}\left(\frac{\pi{s}}{\mathrm{2}}\right)\underset{{m}=\mathrm{1}} {\overset{\infty} {\sum}}\left[\frac{{sin}\left(\mathrm{2}\pi{mx}\right)}{{m}^{\left(\mathrm{1}−{s}\right)} }\right]\right\} \\ $$ Answered by witcher3 last…
Question Number 198166 by mr W last updated on 12/Oct/23 $${if}\:{f}\left({x}\right)={x}^{\mathrm{2}} +{bx}+{c} \\ $$$${f}\left({f}\left(\mathrm{1}\right)\right)={f}\left({f}\left(\mathrm{2}\right)\right)=\mathrm{0}\:{and}\:{f}\left(\mathrm{1}\right)\neq{f}\left(\mathrm{2}\right) \\ $$$${find}\:{f}\left(\mathrm{0}\right)=? \\ $$ Answered by mr W last updated on…