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In-the-expansion-1-1-2-x-n-the-coefficient-of-x-m-is-5-4-times-the-coefficient-of-x-m-1-a-Show-that-5n-13m-8-b-If-m-and-n-are-positive-integers-such-that-m-n-determine-the-smalles

Question Number 154225 by ZiYangLee last updated on 15/Sep/21 $$\mathrm{In}\:\mathrm{the}\:\mathrm{expansion}\:\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}\right)^{{n}} ,\:\mathrm{the}\:\mathrm{coefficient} \\ $$$$\mathrm{of}\:{x}^{{m}} \:\mathrm{is}\:\frac{\mathrm{5}}{\mathrm{4}}\:\mathrm{times}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:{x}^{{m}+\mathrm{1}} . \\ $$$$\left.{a}\right)\:\mathrm{Show}\:\mathrm{that}\:\mathrm{5}{n}−\mathrm{13}{m}=\mathrm{8} \\ $$$$\left.{b}\right)\:\mathrm{If}\:{m}\:\mathrm{and}\:{n}\:\mathrm{are}\:\mathrm{positive}\:\mathrm{integers},\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\:\:\:{m}\leqslant{n},\:\mathrm{determine}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{values}\:\mathrm{of} \\ $$$$\:\:\:\:\:{m}\:\mathrm{and}\:{n}. \\ $$…

Question-154210

Question Number 154210 by Eric002 last updated on 15/Sep/21 Commented by Eric002 last updated on 15/Sep/21 $${the}\:{indentical}\:{circles}\:{of}\:{radius}\:{r},\:{inside} \\ $$$${a}\:{triangle},\:{each}\:{of}\:{them}\:{tangent}\:{to}\:{tow}\: \\ $$$${sides}\:{of}\:{the}\:{triangle}. \\ $$$${prove}:\frac{\mathrm{1}}{{r}}=\frac{\mathrm{1}}{{R}_{{in}} }+\frac{\mathrm{1}}{{R}_{{out}} }…

log-2-x-log-4-y-4-x-y-8-

Question Number 88663 by mary_ last updated on 12/Apr/20 $$\begin{cases}{{log}_{\mathrm{2}} {x}+{log}_{\mathrm{4}} {y}=\mathrm{4}}\\{{x}.{y}=\mathrm{8}}\end{cases} \\ $$ Answered by jagoll last updated on 28/Apr/20 $$\mathrm{log}_{\mathrm{2}} \left(\mathrm{x}\right)+\frac{\mathrm{1}}{\mathrm{2}}\mathrm{log}_{\mathrm{2}} \left(\frac{\mathrm{8}}{\mathrm{x}}\right)\:=\mathrm{4} \\…

chose-the-right-option-1-if-x-y-Q-c-and-x-lt-y-then-a-Q-such-that-x-lt-a-lt-y-in-case-we-say-that-a-Q-c-is-an-ordered-field-b-Q-dense-inQ-c-c-not-ordered-field-2-let-Q-A-Q-A-not-necessary-ha

Question Number 88607 by M±th+et£s last updated on 11/Apr/20 $${chose}\:{the}\:{right}\:{option} \\ $$$$\left.\mathrm{1}\right)\:{if}\:{x},{y}\:\in{Q}^{{c}} \:{and}\:{x}<{y}\:\:{then}\:\exists{a}\in{Q} \\ $$$${such}\:{that}\:{x}<{a}<{y}\:{in}\:{case}\:{we}\:{say}\:{that} \\ $$$$\left.{a}\right){Q}^{{c}} \:{is}\:{an}\:{ordered}\:{field} \\ $$$$\left.{b}\right){Q}\:{dense}\:{inQ}^{{c}} \\ $$$$\left.{c}\right){not}\:{ordered}\:{field} \\ $$$$ \\…

Question-23059

Question Number 23059 by Physics lover last updated on 25/Oct/17 Commented by Physics lover last updated on 25/Oct/17 $${In}\:{the}\:{arrangement}\:{shown}\:{here}\: \\ $$$${friction}\:{between}\:{the}\:{spool}\: \\ $$$$\left({of}\:{mass}\:{m},\:{moment}\:{of}\:{inertia}\:{I}\:\right) \\ $$$${and}\:{the}\:{plank}\:\left({of}\:{mass}\:{M}\right)…