Question Number 139548 by peter frank last updated on 28/Apr/21 Answered by Dwaipayan Shikari last updated on 28/Apr/21 $$\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}} {sin}\left({x}\right)}{\mathrm{1}+{x}^{\mathrm{6}} }{dx}=−\int_{−\frac{\pi}{\mathrm{2}}} ^{\frac{\pi}{\mathrm{2}}} \frac{{x}^{\mathrm{2}}…
Question Number 73974 by Raxreedoroid last updated on 17/Nov/19 $$\mathrm{Prove}\:\mathrm{this}\:\mathrm{equation} \\ $$$$\frac{\mathrm{2}^{{x}−\mathrm{1}} \left({x}−\frac{\mathrm{1}}{\mathrm{2}}\right)!}{\frac{\mathrm{1}}{\mathrm{2}}!}=\frac{\left(\mathrm{2}{x}−\mathrm{1}\right)!}{\mathrm{2}^{{x}−\mathrm{1}} \left({x}−\mathrm{1}\right)!} \\ $$ Answered by mind is power last updated on 17/Nov/19…
Question Number 73818 by rajesh4661kumar@gmail.com last updated on 16/Nov/19 Commented by Tinku Tara last updated on 16/Nov/19 $$\mathrm{Can}\:\mathrm{you}\:\mathrm{post}\:\mathrm{a}\:\mathrm{clear}\:\mathrm{image}. \\ $$ Answered by Tanmay chaudhury last…
Question Number 8252 by 8168 last updated on 04/Oct/16 Answered by Yozzias last updated on 04/Oct/16 $$\mathrm{6}+\mathrm{4}+\frac{\mathrm{8}}{\mathrm{3}}+\frac{\mathrm{16}}{\mathrm{9}}+… \\ $$$$=\mathrm{6}+\left(\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}^{\mathrm{0}} }+\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{3}^{\mathrm{1}} }+\frac{\mathrm{2}^{\mathrm{4}} }{\mathrm{3}^{\mathrm{2}} }+…+\frac{\mathrm{2}^{\mathrm{n}+\mathrm{1}}…
Question Number 8232 by FilupSmith last updated on 03/Oct/16 $$\mathrm{Every}\:\mathrm{day},\:\mathrm{for}\:{n}\:\mathrm{days},\:\mathrm{you}\:\mathrm{put}\:\mathrm{either} \\ $$$$\$\mathrm{1},\:\$\mathrm{2},\:\mathrm{or}\:\$\mathrm{3}\:\mathrm{into}\:\mathrm{a}\:\mathrm{saving}\:\mathrm{account}. \\ $$$$\mathrm{It}\:\mathrm{is}\:\mathrm{random}\:\mathrm{as}\:\mathrm{to}\:\mathrm{how}\:\mathrm{much}\:\mathrm{you}\:\mathrm{save} \\ $$$$\mathrm{each}\:\mathrm{day}.\:\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:{average}\:\mathrm{amount} \\ $$$$\mathrm{you}\:\mathrm{will}\:\mathrm{have}\:\mathrm{saved}\:\mathrm{in}\:{n}\:\mathrm{days}? \\ $$ Answered by prakash jain last…
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Question Number 138989 by metamorfose last updated on 20/Apr/21 $${find}\:{the}\:{integers}\:{x}\:,\:{y}\:,\:{z}\:,\:{n}\:{that} \\ $$$${satisfy}\::\:\mathrm{2}^{{n}} ={x}!+{y}!+{z}! \\ $$ Answered by mindispower last updated on 21/Apr/21 $${let}\:{m}={min}\left({x},{y},{z}\right),{if}\:{m}\geqslant\mathrm{3}\Rightarrow \\ $$$${x}!+{y}!+{z}!\equiv\mathrm{0}\left[\mathrm{3}\right]\Leftrightarrow\mathrm{2}^{{n}}…
Question Number 7881 by Rasheed Soomro last updated on 23/Sep/16 $${A}\:{gets}\:\mathrm{1}/\mathrm{5}\:\:{of}\:\:{some}\:\:{amount}, \\ $$$${B}\:\:{gets}\:\mathrm{1}/\mathrm{3}\:\:{of}\:\:{remaining},\:{C} \\ $$$${gets}\:\mathrm{1}/\mathrm{6}\:{of}\:{remaining},\:{D}\:{gets} \\ $$$$\mathrm{1}/\mathrm{7}\:\:{of}\:\:{remaining}\:\:{and}\:{E}\:{gets} \\ $$$${rest}\:{of}\:{amount}.\:{What}\:{fraction} \\ $$$${gets}\:{E}\:? \\ $$ Commented by…
Question Number 73340 by Raxreedoroid last updated on 10/Nov/19 $$\mathrm{Is}\:\mathrm{there}\:\mathrm{any}\:\mathrm{pi}/\mathrm{product}\:\mathrm{notation}\:\mathrm{rules} \\ $$$$\mathrm{I}\:\mathrm{discovered}\:\mathrm{some}\:\mathrm{such}\:\mathrm{as}: \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{k}\right]=\frac{{b}!}{\left({a}−\mathrm{1}\right)!} \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{c}\right]={c}^{{b}−{a}+\mathrm{1}} \\ $$$$\underset{{k}={a}} {\overset{{b}} {\prod}}\left[{c}\centerdot{k}\right]=\underset{{k}={a}} {\overset{{b}}…
Question Number 7771 by FilupSmith last updated on 14/Sep/16 $$\mathrm{Show}\:\mathrm{that}: \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}}{\left({n}+\mathrm{2}\right)!}=\mathrm{3}−{e} \\ $$ Commented by FilupSmith last updated on 14/Sep/16 $$\mathrm{Amazing}! \\…