Category: Algebra

Question Number 208218 by hardmath last updated on 07/Jun/24 $${\mathrm{a}}_{\mathbf{n}}\mathrm{numbers}\mathrm{series}$$$$\mathrm{If}{\mathrm{S}}_{16}-{\mathrm{S}}_{13}={\mathrm{S}}_{106}-{\mathrm{S}}_{103}$$$$\mathrm{Find}:\frac{{3\mathrm{a}}_3+{4\mathrm{a}}_4+{5\mathrm{a}}_5}{{2\mathrm{a}}_{12}}=?$$ Commented…

Question Number 208199 by efronzo1 last updated on 07/Jun/24 Answered by Frix last updated on 07/Jun/24 $$y=\sqrt{18+3x-x^2}\mathrm{is}\mathrm{a}\mathrm{semi}\mathrm{circle}\mathrm{with}r=\frac{9}{2}$$$$\sqrt{x+3}+\sqrt{6-x}\mathrm{has}\mathrm{the}\mathrm{maximum}\left(\begin{array}{c}\frac{3}{2}\\ 3\sqrt{2}\end{array}\right)$$$$\mathrm{We}\mathrm{have}2\mathrm{solutions}\mathrm{for}0⩽m<\frac{3\sqrt{2}}{4}\mathrm{and}$$$$\mathrm{exactly}\:\mathrm{one}\:\mathrm{solution}\:\mathrm{at}\:{m}=\frac{\mathrm{3}\sqrt{\mathrm{2}}}{\mathrm{4}} \…

Question Number 208164 by Fridunatjan08 last updated on 06/Jun/24 $$Solveforx:$$$${x}^{\mathrm{2}} +{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)+{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{2}} +{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{3}} +…+{x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{\mathrm{100}} =\mathrm{1} \…

Question Number 208167 by hardmath last updated on 06/Jun/24 $$\mathrm{y}=3{\cos }^2\alpha +2\cos \alpha$$$$\mathrm{find}:\max \left(\mathrm{y}\right)=?$$ Answered by A5T last updated on 07/Jun/24 $$y⩽3\times 1+2\times 1=5\Rightarrow max\left(y\right)=5(Equalityat\alpha =0]$$…

Question Number 208097 by efronzo1 last updated on 05/Jun/24 $$\downharpoonleft \underset{―}{}$$ Answered by MM42 last updated on 05/Jun/24 $$2^{x-m+1}-2^{x-m}=8$$$$\Rightarrow\mathrm{2}^{{x}−{m}} =\mathrm{8}\Rightarrow{x}−{m}=\mathrm{3}\Rightarrow{x}={m}+\mathrm{3}…