Category: Arithmetic

Question Number 103832 by bemath last updated on 17/Jul/20 $$p\left(x\right)=x^4+{ax}^3+{bx}^2+cx+d$$$$ifp\left(1\right)=10,p\left(2\right)=20and$$$$p\left(3\right)=30.find\frac{p\left(12\right)+p(-8)}{10}$$ Answered by floor(10²Eta[1]) last updated on…

Question Number 103780 by bobhans last updated on 17/Jul/20 $$thethird,sixthandseventhtermsofa$$$$geometricprogression(whosecommon$$$$ratioisneither0nor1)arein$$$$arithmeticprogression.provethatthe$$$$sumofthefirstthreetermsisequalto$$$$fourth.$$ Answered by bemath…

Question Number 103700 by anonymous last updated on 16/Jul/20 Commented by anonymous last updated on 16/Jul/20 $$plzzhelpme$$ Answered by bobhans last updated on…

Question Number 103673 by bobhans last updated on 16/Jul/20 $$\underset{k=1}{\sum ^{4095}}\frac{1}{(\sqrt{k}+\sqrt{k+1})(\sqrt[4]{k}+\sqrt[4]{k+1})}?$$ Answered by bramlex last updated on 16/Jul/20 $$\frac{1}{(\sqrt{\mathrm{k}}+\sqrt{\mathrm{k}+1})(\sqrt[4]{\mathrm{k}}+\sqrt[4]{\mathrm{k}+1})}\times \frac{\sqrt[4]{\mathrm{k}+1}-\sqrt[4]{\mathrm{k}}}{\sqrt[4]{\mathrm{k}+1}-\sqrt[4]{\mathrm{k}}}=$$$$\frac{\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{\mathrm{k}}}{\left(\sqrt{\mathrm{k}+\mathrm{1}}+\sqrt{\mathrm{k}}\right)\left(\sqrt{\mathrm{k}+\mathrm{1}}−\sqrt{\mathrm{k}}\right)}\:=\:\sqrt[{\mathrm{4}}]{\mathrm{k}+\mathrm{1}}−\sqrt[{\mathrm{4}}]{\mathrm{k}} \…

Question Number 103648 by Lordose last updated on 16/Jul/20 $$\mathbf{Evaluate}\frac{1}{1\cdot 2\cdot 3}+\frac{3}{2\cdot 3\cdot 4}+\frac{5}{3\cdot 4\cdot 5}+\dots +\frac{2\mathbf{n}-1}{\mathbf{n}(\mathbf{n}+1)(\mathbf{n}+2}$$ Answered by Dwaipayan Shikari last updated on 16/Jul/20 $$T_n=\frac{2n-1}{n(n+1)(n+2)}$$$$\Sigma{T}_{{n}} =\Sigma\frac{{n}+{n}+\mathrm{2}−\mathrm{3}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}=\Sigma\frac{\mathrm{1}}{{n}\left({n}+\mathrm{1}\right)}+\Sigma\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}−\frac{\mathrm{3}}{\mathrm{2}}\Sigma\frac{\mathrm{2}+{n}−{n}}{{n}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}…