Question Number 153916 by EDWIN88 last updated on 12/Sep/21 $${The}\:{value}\:{of}\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\:\frac{\left(\mathrm{3}_{{n}} \right)\left(\mathrm{2}_{{n}} \right){x}^{{n}} }{\left(\mathrm{1}_{{n}} \right){n}!}\:\beta\left(\mathrm{2},{n}+\mathrm{1}\right)\:{is} \\ $$$${a}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\mathrm{2}_{{n}} \right)\frac{{x}^{{n}} }{{n}!} \\ $$$${b}.\:\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty}…
Question Number 88261 by ar247 last updated on 09/Apr/20 Answered by Joel578 last updated on 09/Apr/20 $${u}\:\equiv\:{v}\:\left(\mathrm{mod}\:{m}\right)\:\Rightarrow\:{u}\:−\:{v}\:=\:{mx},\:{x}\:\in\:\mathbb{Z} \\ $$$$\left(\mathrm{1}\right)\:{u}\:=\:{mx}\:+\:{v} \\ $$$$\mathrm{Since}\:\mathrm{gcf}\left({v},{m}\right)\:\mathrm{divides}\:\mathrm{both}\:{v}\:\mathrm{and}\:\mathrm{m},\:\mathrm{it}\:\mathrm{also}\:\mathrm{divides}\:{u}, \\ $$$$\mathrm{hence}\:\mathrm{gcf}\left({v},{m}\right)\:\mathrm{divides}\:\mathrm{gcf}\left({u},{m}\right) \\ $$$$\Rightarrow\:\mathrm{gcf}\left({v},{m}\right)\:\mid\:\mathrm{gcf}\left({u},{m}\right)…
Question Number 22635 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n},\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{20}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{2}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} . \\ $$$$ \\…
Question Number 22625 by Rasheed.Sindhi last updated on 21/Oct/17 $$\mathrm{For}\:\mathrm{each}\:\mathrm{positive}\:\mathrm{integer}\:\mathrm{n}\:\mathrm{define} \\ $$$$\mathrm{a}_{\mathrm{n}} =\mathrm{30}+\mathrm{n}^{\mathrm{2}} ,\mathrm{and}\:\mathrm{d}_{\mathrm{n}} =\mathrm{gcd}\left(\mathrm{a}_{\mathrm{n}} ,\mathrm{a}_{\mathrm{n}+\mathrm{1}} \right). \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:\mathrm{values}\:\mathrm{that}\:\mathrm{are} \\ $$$$\mathrm{taken}\:\mathrm{by}\:\mathrm{d}_{\mathrm{n}} \:\mathrm{and}\:\mathrm{show}\:\mathrm{by}\:\mathrm{examples} \\ $$$$\mathrm{that}\:\mathrm{each}\:\mathrm{of}\:\mathrm{these}\:\mathrm{values}\:\mathrm{are}\:\mathrm{attained}. \\…
Question Number 88050 by Sahil vampire last updated on 08/Apr/20 Commented by mr W last updated on 08/Apr/20 $${i}\:{found}\:{there}\:{is}\:{only}\:{one}\:{such}\:{number}: \\ $$$$\mathrm{588}\:\mathrm{2353} \\ $$$$\mathrm{588}^{\mathrm{2}} +\mathrm{2353}^{\mathrm{2}} =\mathrm{588}\:\mathrm{2353}…
Question Number 88000 by jdmath last updated on 07/Apr/20 $${Is}\:{there}\:{a}\:{formula}\:{to}\:{calculate}\: \\ $$$$ \\ $$$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{i}^{\mathrm{2}} } \\ $$$${interms}\:{of}\:{n}..? \\ $$ Commented by abdomathmax last…
Question Number 153458 by liberty last updated on 07/Sep/21 $${Given}\:{a}\:{set}\:{consisting}\:{of}\:\mathrm{22}\:{integer} \\ $$$$\:{A}=\left\{\pm{a}_{\mathrm{1}} ,\pm{a}_{\mathrm{2}} ,…,\pm{a}_{\mathrm{11}} \right\}.\:{Show}\:{that} \\ $$$${exist}\:{subset}\:{of}\:{S}\:{with}\:{properties} \\ $$$$\left(\mathrm{1}\right)\:{for}\:{every}\:{i}=\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{11}\: \\ $$$$\:{have}\:{least}\:{one}\:{between}\:{a}_{{i}} \:{or}\:−{a}_{{i}} \\ $$$$\:{element}\:{of}\:{S} \\…
Question Number 22372 by Rasheed.Sindhi last updated on 16/Oct/17 Commented by Rasheed.Sindhi last updated on 16/Oct/17 $$\mathrm{Question}\:\mathrm{asked}\:\mathrm{by}\:\mathrm{Mr}.\:\mathrm{Tinkutara}. \\ $$$$\mathrm{reposted}\:\mathrm{for}\:\mathrm{answer}. \\ $$ Commented by Tinkutara last…
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Question Number 22080 by Tinkutara last updated on 10/Oct/17 $$\mathrm{Given}\:\mathrm{any}\:\mathrm{positive}\:\mathrm{integer}\:{n}\:\mathrm{show} \\ $$$$\mathrm{that}\:\mathrm{there}\:\mathrm{are}\:\mathrm{two}\:\mathrm{positive}\:\mathrm{rational} \\ $$$$\mathrm{numbers}\:{a}\:\mathrm{and}\:{b},\:{a}\:\neq\:{b},\:\mathrm{which}\:\mathrm{are}\:\mathrm{not} \\ $$$$\mathrm{integers}\:\mathrm{and}\:\mathrm{which}\:\mathrm{are}\:\mathrm{such}\:\mathrm{that}\:{a}\:−\:{b}, \\ $$$${a}^{\mathrm{2}} \:−\:{b}^{\mathrm{2}} ,\:{a}^{\mathrm{3}} \:−\:{b}^{\mathrm{3}} ,\:…..,\:{a}^{{n}} \:−\:{b}^{{n}} \:\mathrm{are}\:\mathrm{all} \\…