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Question-163318

Question Number 163318 by KONE last updated on 06/Jan/22 Answered by Mathspace last updated on 06/Jan/22 $$={lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{{n}}\sum_{{k}=\mathrm{1}} ^{{n}} \left[{e}^{\frac{{k}}{{n}}} \right] \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left[{e}^{{x}}…

Question-32241

Question Number 32241 by mondodotto@gmail.com last updated on 22/Mar/18 Commented by MJS last updated on 22/Mar/18 $$\mathrm{I}\:\mathrm{don}'\mathrm{t}\:\mathrm{think}\:\mathrm{that}'\mathrm{s}\:\mathrm{true}\:\mathrm{if}\:\mathrm{you} \\ $$$$\mathrm{can}\:\mathrm{freely}\:\mathrm{choose}\:{a},\:{b}\:\mathrm{and}\:\theta \\ $$$$\mathrm{but}\:\mathrm{of}\:\mathrm{course}\:\mathrm{you}\:\mathrm{can}\:\mathrm{find}\:\theta \\ $$$$\mathrm{for}\:\mathrm{any}\:\mathrm{given}\:\mathrm{triplet}\:{a},\:{b},\:{c}\:\:\mathrm{with} \\ $$$$\mathrm{or}\:\mathrm{without}\:{a}^{\mathrm{2}}…

Question-32240

Question Number 32240 by mondodotto@gmail.com last updated on 22/Mar/18 Answered by MJS last updated on 22/Mar/18 $$\frac{\mathrm{cos}\:\alpha}{\mathrm{sin}\:\beta×\mathrm{sin}\:\gamma}+\frac{\mathrm{cos}\:\beta}{\mathrm{sin}\:\alpha×\mathrm{sin}\:\gamma}+\frac{\mathrm{cos}\:\gamma}{\mathrm{sin}\:\alpha×\mathrm{sin}\:\beta}= \\ $$$$=\frac{\mathrm{cos}\:\alpha×\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta×\mathrm{sin}\:\beta+\mathrm{cos}\:\gamma×\mathrm{sin}\:\gamma}{\mathrm{sin}\:\alpha×\mathrm{sin}\:\beta×\mathrm{sin}\:\gamma} \\ $$$$ \\ $$$$\mathrm{I}. \\ $$$$\mathrm{cos}\:\alpha×\mathrm{sin}\:\alpha+\mathrm{cos}\:\beta×\mathrm{sin}\:\beta+\mathrm{cos}\:\gamma×\mathrm{sin}\:\gamma=…

Question-163259

Question Number 163259 by SANOGO last updated on 05/Jan/22 Answered by TheSupreme last updated on 05/Jan/22 $$\left.\mathrm{1}\right)\: \\ $$$${b}^{{n}} −\mathrm{1}\equiv\mathrm{0}\left({b}−\mathrm{1}\right) \\ $$$$\left({b}−\mathrm{1}\right){P}_{{n}−\mathrm{1}} \left({b}\right)={b}^{{n}} −\mathrm{1} \\…