Question Number 73040 by mathmax by abdo last updated on 05/Nov/19 $${prove}\:{that}\:\:\forall\left({n},{p}\right)\in{N}^{\bigstar} ×{N} \\ $$$$\left.\mathrm{1}\right)\sum_{{k}=\mathrm{0}} ^{{p}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$$$\left.\mathrm{2}\right)\forall\left({p},{q}\right)\in{N}^{\mathrm{2}} \:\:\:\:\sum_{{k}=\mathrm{0}}…
Question Number 73041 by mathmax by abdo last updated on 05/Nov/19 $${prove}\:{that}\:\forall{n}\in\:{N}\:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\left(−\mathrm{1}\right)^{{k}} \:\left({C}_{\mathrm{2}{n}} ^{{k}} \right)^{\mathrm{2}} \:=\left(−\mathrm{1}\right)^{{n}} \:{C}_{\mathrm{2}{n}} ^{{n}} \\ $$ Commented by mathmax…
Question Number 73039 by mathmax by abdo last updated on 05/Nov/19 $${let}\:{U}_{{n}} =\frac{{n}}{\mathrm{2}}\:{if}\:{n}\:{even}\:{and}\:{U}_{{n}} =\frac{{n}−\mathrm{1}}{\mathrm{2}}\:{if}\:{n}\:{odd}\:{let}\:{f}\left({n}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {U}_{{k}} \\ $$$${prove}\:{that}\:\forall\left({x},{y}\right)\in{N}^{\mathrm{2}} \:\:\:\:{f}\left({x}+{y}\right)−{f}\left({x}−{y}\right)={xy} \\ $$ Answered by mind is…
Question Number 138575 by KwesiDerek last updated on 15/Apr/21 $$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{2}^{\boldsymbol{\mathrm{x}}} =\mathrm{1} \\ $$$$\boldsymbol{\mathrm{Any}}\:\boldsymbol{\mathrm{help}} \\ $$ Commented by soudo last updated on 15/Apr/21…
Question Number 73036 by mathmax by abdo last updated on 05/Nov/19 $$\left.{calculate}\:\mathrm{1}\right)\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{2}} \left({n}+\mathrm{1}−{k}\right) \\ $$$$\left.\mathrm{2}\right)\sum_{\mathrm{1}\leqslant{i}\leqslant{j}\leqslant{n}} \:{ij} \\ $$ Commented by mathmax by abdo…
Question Number 7501 by Yozzia last updated on 01/Sep/16 $${Compute}\:\underset{{A}\rightarrow+\infty} {\mathrm{lim}}\left\{\frac{\mathrm{1}}{{A}}\int_{\mathrm{1}} ^{{A}} {A}^{\frac{\mathrm{1}}{{x}}} {dx}\right\}. \\ $$$$\left({IMC}\:\mathrm{2015}\right) \\ $$ Answered by FilupSmith last updated on 01/Sep/16…
Question Number 73037 by aliesam last updated on 05/Nov/19 Commented by mathmax by abdo last updated on 06/Nov/19 $$\left.\mathrm{1}\right)\:{I}_{\mathrm{0}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{e}^{\mathrm{1}−{x}} {dx}\:=\left[−{e}^{\mathrm{1}−{x}} \right]_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 7500 by Tawakalitu. last updated on 31/Aug/16 $$\mathrm{5}^{\sqrt{{x}\:}} \:−\:\:\mathrm{5}^{{x}\:−\:\mathrm{7}} \:\:=\:\:\mathrm{100} \\ $$$$ \\ $$$${Find}\:{the}\:{value}\:{of}\:{x}. \\ $$$${x}\:=\:\mathrm{9}\:\:\:{please}\:{workings}. \\ $$ Answered by sandy_suhendra last updated…
Question Number 138569 by I want to learn more last updated on 14/Apr/21 Answered by mr W last updated on 15/Apr/21 $$\left({a}\right) \\ $$$$\mathrm{2}{b}×{c}×\frac{{c}}{\mathrm{2}}={a}×{b}×\frac{{b}}{\mathrm{2}} \\…
Question Number 73034 by mathmax by abdo last updated on 05/Nov/19 $${calculate}\:{U}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{{k}}{\left({k}+\mathrm{1}\right)!} \\ $$ Commented by mathmax by abdo last updated on…