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Category: Algebra

Given-only-the-standard-result-r-1-N-r-2-1-6-N-N-1-2N-1-is-applied-to-determining-the-series-1-2-2-2-2-3-2-2-4-2-5-2-2-6-2-7-2-2-n-1-2-n-2-where-n-is-odd-show-that-it-is-given

Question Number 669 by 112358 last updated on 21/Feb/15 $${Given}\:{only}\:{the}\:{standard}\:{result}\:\underset{{r}=\mathrm{1}} {\overset{{N}} {\sum}}{r}^{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{6}}{N}\left({N}+\mathrm{1}\right)\left(\mathrm{2}{N}+\mathrm{1}\right)\: \\ $$$${is}\:{applied}\:{to}\:{determining}\:{the}\:{series} \\ $$$$\mathrm{1}^{\mathrm{2}} +\mathrm{2}×\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{2}×\mathrm{4}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{2}×\mathrm{6}^{\mathrm{2}} +\mathrm{7}^{\mathrm{2}} +…+\mathrm{2}\left({n}−\mathrm{1}\right)^{\mathrm{2}} +{n}^{\mathrm{2}}…

log-x-y-pi-log-y-x-e-a-1-log-y-x-pi-1-1-log-x-y-e-1-b-x-a-b-2e-y-a-b-2pi-

Question Number 647 by 123456 last updated on 18/Feb/15 $$\mathrm{log}_{{x}} \left({y}^{\pi} \right)+\mathrm{log}_{{y}} \left({x}^{{e}} \right)={a} \\ $$$$\frac{\mathrm{1}}{\mathrm{log}_{{y}} \left({x}^{\pi^{−\mathrm{1}} } \right)}−\frac{\mathrm{1}}{\mathrm{log}_{{x}} \left({y}^{{e}^{−\mathrm{1}} } \right)}={b} \\ $$$$\frac{{x}^{{a}+{b}+\mathrm{2}{e}} }{{y}^{{a}−{b}+\mathrm{2}\pi}…

x-x-6-x-

Question Number 131697 by mathlove last updated on 07/Feb/21 $${x}^{{x}} =\mathrm{6}\:\:\:\:\:\:\:\:\:\:{x}=? \\ $$ Answered by mr W last updated on 07/Feb/21 $${x}=\mathrm{6}^{\frac{\mathrm{1}}{{x}}} ={e}^{\frac{\mathrm{ln}\:\mathrm{6}}{{x}}} \\ $$$$\frac{\mathrm{1}}{{x}}{e}^{\frac{\mathrm{ln}\:\mathrm{6}}{{x}}}…

2-3-2-3-

Question Number 619 by Cheenz last updated on 11/Feb/15 $$\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\:+\:\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}\:=\:? \\ $$ Answered by 123456 last updated on 11/Feb/15 $${x}=\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}+\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}} \\ $$$${x}^{\mathrm{2}} =\mathrm{2}+\sqrt{\mathrm{3}}+\mathrm{2}\sqrt{\mathrm{2}+\sqrt{\mathrm{3}}}\sqrt{\mathrm{2}−\sqrt{\mathrm{3}}}+\mathrm{2}−\sqrt{\mathrm{3}} \\ $$$${x}^{\mathrm{2}}…

Let-and-are-the-roots-of-the-equation-x-2-6x-2-0-If-a-n-n-n-for-n-1-then-the-value-of-a-10-2a-8-2a-9-

Question Number 131688 by liberty last updated on 07/Feb/21 $$\:\mathrm{Let}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\: \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} −\mathrm{6x}−\mathrm{2}=\mathrm{0}. \\ $$$$\mathrm{If}\:{a}_{{n}} \:=\:\alpha^{{n}} −\beta^{{n}} \:\mathrm{for}\:{n}\:\geqslant\mathrm{1}\: \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\frac{{a}_{\mathrm{10}} −\mathrm{2}{a}_{\mathrm{8}} }{\mathrm{2}{a}_{\mathrm{9}} }\:? \\ $$…

Is-there-any-formula-to-find-sum-of-1-n-2-n-4-n-6-n-8-n-2k-where-n-k-Z-

Question Number 66126 by Joel122 last updated on 09/Aug/19 $$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{formula}\:\mathrm{to}\:\mathrm{find}\:\mathrm{sum}\:\mathrm{of} \\ $$$$\mathrm{1}\:+\:{n}^{\mathrm{2}} \:+\:{n}^{\mathrm{4}} \:+\:{n}^{\mathrm{6}} \:+\:{n}^{\mathrm{8}} \:+\:…\:+\:{n}^{\mathrm{2}{k}} \:+\:… \\ $$$$\mathrm{where}\:{n},{k}\:\in\:\mathbb{Z}^{+} \: \\ $$ Answered by mr…

a-b-10-a-c-24-b-c-20-find-a-b-c-

Question Number 588 by shaleen last updated on 03/Feb/15 $$\left({a}+{b}=\mathrm{10}\right)\:\left({a}+{c}=\mathrm{24}\right)\:\left({b}+{c}=\mathrm{20}\right) \\ $$$${find}\:{a}\:{b}\:{c} \\ $$ Answered by prakash jain last updated on 03/Feb/15 $${a}+{b}=\mathrm{10}\Rightarrow{b}=\mathrm{10}−{a} \\ $$$${a}+{c}=\mathrm{24}\Rightarrow{c}=\mathrm{24}−{a}…