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Category: Arithmetic

Given-a-function-x-1-f-x-1-x-4x-f-1-x-100-x-2-4-x-x-0-x-1-Find-f-2-f-3-f-4-f-400-

Question Number 166515 by greogoury55 last updated on 21/Feb/22 $$\:\:\:\:{Given}\:{a}\:{function}\: \\ $$$$\:\:\:\left({x}+\mathrm{1}\right){f}\left(−{x}\right)+\frac{\mathrm{1}−{x}}{\mathrm{4}{x}}\:{f}\left(\frac{\mathrm{1}}{{x}}\right)=\frac{\mathrm{100}\left({x}^{\mathrm{2}} +\mathrm{4}\right)}{{x}} \\ $$$$\:{x}\neq\mathrm{0}\:,\:{x}\neq\mathrm{1} \\ $$$$\:{Find}\:{f}\left(\mathrm{2}\right)+{f}\left(\mathrm{3}\right)+{f}\left(\mathrm{4}\right)+…+{f}\left(\mathrm{400}\right) \\ $$ Answered by nurtani last updated on…

Consider-the-sequences-u-n-and-v-n-defined-by-u-0-1-u-n-1-2u-n-v-n-u-n-v-n-and-v-0-2-v-n-1-u-n-v-n-2-n-N-1-Show-that-u-n-

Question Number 100769 by Rio Michael last updated on 28/Jun/20 $$\mathrm{Consider}\:\mathrm{the}\:\mathrm{sequences}\:\left({u}_{{n}} \right)\:\mathrm{and}\:\left({v}_{{n}} \right)\:\mathrm{defined}\:\mathrm{by} \\ $$$$\:\begin{cases}{{u}_{\mathrm{0}} \:=\:\mathrm{1}}\\{{u}_{{n}+\mathrm{1}} \:=\:\frac{\mathrm{2}{u}_{{n}} {v}_{{n}} }{{u}_{{n}} \:+\:{v}_{{n}} }}\end{cases}\:\mathrm{and}\:\begin{cases}{{v}_{\mathrm{0}} \:=\:\mathrm{2}}\\{{v}_{{n}+\mathrm{1}} \:=\:\frac{{u}_{{n}} \:+\:{v}_{{n}} }{\mathrm{2}}}\end{cases}\:\:\forall\:{n}\in\:\mathbb{N}…

sketch-x-2-y-3-and-x-2-y-2-16-and-hence-solve-x-2-y-3-x-2-y-2-16-0-

Question Number 100767 by Rio Michael last updated on 28/Jun/20 $$\:\mathrm{sketch}\:\:{x}^{\mathrm{2}} \:=\:{y}^{\mathrm{3}} \:\mathrm{and}\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:=\:\mathrm{16} \\ $$$$\mathrm{and}\:\mathrm{hence}\:\mathrm{solve}\:\left({x}^{\mathrm{2}} −{y}^{\mathrm{3}} \right)\left({x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} −\mathrm{16}\right)\:\geqslant\mathrm{0} \\ $$ Answered by…

Question-35150

Question Number 35150 by Victor31926 last updated on 16/May/18 Answered by Rasheed.Sindhi last updated on 16/May/18 $$\bullet\mathrm{Any}\:\mathrm{power}\:\mathrm{of}\:\mathrm{even}\:\mathrm{number}\:\mathrm{is}\:\mathrm{even} \\ $$$$\:\:\:\:\:\mathrm{and}\:\mathrm{even}\:\mathrm{number}\:\mathrm{gives}\:\mathrm{remainder}\:\mathrm{0} \\ $$$$\:\:\:\:\:\mathrm{on}\:\mathrm{dividing}\:\mathrm{by}\:\mathrm{2} \\ $$$$\bullet\mathrm{Any}\:\mathrm{power}\:\mathrm{of}\:\mathrm{odd}\:\mathrm{number}\:\mathrm{is}\:\mathrm{odd} \\ $$$$\:\:\:\:\:\mathrm{and}\:\mathrm{odd}\:\mathrm{number}\:\mathrm{gives}\:\mathrm{remainder}\:\mathrm{1}…

let-U-n-be-a-sequence-definied-by-U-0-1-U-n-1-3U-n-2-U-n-2-show-that-0-lt-U-n-lt-2-

Question Number 100640 by pticantor last updated on 27/Jun/20 $${let}\left(\:\boldsymbol{{U}}_{{n}} \right)\:{be}\:{a}\:{sequence}\:{definied}\:{by}: \\ $$$$\begin{cases}{\boldsymbol{{U}}_{\mathrm{0}} =\mathrm{1}}\\{\boldsymbol{{U}}_{{n}+\mathrm{1}} =\frac{\mathrm{3}\boldsymbol{{U}}_{\boldsymbol{{n}}} +\mathrm{2}}{\boldsymbol{{U}}_{\boldsymbol{{n}}} +\mathrm{2}}}\end{cases} \\ $$$$\boldsymbol{{show}}\:\boldsymbol{{that}}\:\mathrm{0}<\boldsymbol{{U}}_{\boldsymbol{{n}}} <\mathrm{2} \\ $$ Answered by maths…