Category: Algebra

Question Number 219025 by universe last updated on 18/Apr/25 Answered by vnm last updated on 19/Apr/25 $$thegeneratingfunctionof$$$$thissequenceis$$$$f\left(x\right)=\frac{x-2\ln (1-x)+\frac{1}{1-x}-1}{{(2-x)}^2}$$ Answered…

Question Number 218997 by hardmath last updated on 17/Apr/25 Answered by MrGaster last updated on 19/Apr/25 $$4!\sum _{\alpha +\beta +\gamma +\delta =4}\frac{{\left[x\right]}^{\alpha }{\left[x/2\right]}^{\beta }{\left[x/4\right]}^{\gamma }{\left[x/8\right]}^{\delta }}{\alpha !\beta !\gamma !\delta !}={(\left[x\right]+\left[\frac{x}{2}\right]+\left[\frac{x}{4}\right]+\left[\frac{x}{8}\right])}^4$$$$\Rightarrow\left(\left[{x}\right]+\left[\frac{{x}}{\mathrm{2}}\right]+\left[\frac{{x}}{\mathrm{4}}\right]\left[\frac{{x}}{\mathrm{8}}\right]\right)^{\mathrm{4}} \equiv\mathrm{1}\:\left(\mathrm{mod}\:\mathrm{5}\right)\:\left(^{\ast}…

Question Number 218854 by Rojarani last updated on 16/Apr/25 Answered by Hamada1969 last updated on 16/Apr/25 $$k=15127$$$$$$ Commented by Rojarani last…

Question Number 218855 by Lekhraj last updated on 16/Apr/25 $$\sqrt{3}{x}^2=\sqrt{(64-x^2)(x^2-4)}+\sqrt{(81-x^2)(x^2-1)}+\sqrt{(49-x^2)(x^2-1)}\mathit{s}\mathit{o}\mathit{l}\mathit{v}\mathit{e}$$ Terms of Service Privacy Policy Contact:…

Question Number 218850 by hardmath last updated on 16/Apr/25 Answered by MrGaster last updated on 17/Apr/25 $${T}_{{n}} =\frac{\mathrm{1}}{\mathrm{ln}\left(\underset{{r}=\mathrm{1}} {\overset{{n}} {\prod}}\frac{{r}^{\mathrm{2}} }{\mathrm{1}+{r}^{\mathrm{2}} }\right)}=\frac{\mathrm{1}}{\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{ln}\left(\frac{{r}^{\mathrm{2}} }{\mathrm{1}+{r}^{\mathrm{2}}…