Question Number 56597 by Tawa1 last updated on 19/Mar/19 Commented by Tawa1 last updated on 19/Mar/19 $$\mathrm{Please}\:\mathrm{help}\:\mathrm{me}\:\mathrm{prove}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{an}\:\mathrm{arithmetico}\:−\:\mathrm{geometric}\:\mathrm{series}, \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{to}\:\mathrm{infinity}.\:\:\: \\ $$$$\boldsymbol{\mathrm{Again}} \\ $$$$\:\:\:\:\mathrm{How}\:\mathrm{does}\:\mathrm{the}\:\mathrm{3x}\:\mathrm{in}\:\mathrm{the}\:\mathrm{above}\:\mathrm{solution}\:\mathrm{disappear}. \\ $$$$\mathrm{From}\:\:\:\:\:\:\:\mathrm{3x}\:+\:\left(\mathrm{2x}^{\mathrm{2}}…
Question Number 56594 by Joel578 last updated on 19/Mar/19 $$\mathrm{Find}\:\mathrm{the}\:{n}\mathrm{th}\:\mathrm{term}\:\mathrm{of}\:\mathrm{the}\:\mathrm{sequence} \\ $$$$\left({a}\right)\:\frac{\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{1}}{\mathrm{4}},\:\frac{\mathrm{1}}{\mathrm{8}},\:\frac{\mathrm{7}}{\mathrm{62}},\:… \\ $$$$\left({b}\right)\:\frac{\mathrm{1}}{\mathrm{2}},\:\frac{\mathrm{1}}{\mathrm{4}},\:\frac{\mathrm{1}}{\mathrm{8}},\:\mathrm{0},\:… \\ $$ Answered by MJS last updated on 19/Mar/19 $$\mathrm{there}\:\mathrm{are}\:\mathrm{always}\:\infty\:\mathrm{possibilities} \\…
Question Number 187619 by BaliramKumar last updated on 19/Feb/23 $$\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } \:\equiv\:{r}\:\left({mod}\:\:\:\mathrm{7}\right) \\ $$$${r}\:=\:? \\ $$ Answered by SEKRET last updated on 19/Feb/23 $$\left(\mathrm{32}\right)^{\mathrm{32}^{\mathrm{32}} }…
Question Number 122036 by physicstutes last updated on 13/Nov/20 $$\mathrm{Let}\:\:{f}\left({x}\right)\:=\:\left(\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}\right)\:\mathrm{and}\:\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\left[{f}\left({x}\right)\right]^{{r}} \:\mathrm{be}\:\mathrm{a}\:\mathrm{convergent}\:\mathrm{series} \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{x}\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{{x}+\mathrm{1}}{{x}+\mathrm{2}}\right)^{{r}} \right]^{{n}} \:=\:\mathrm{4}\: \\ $$…
Question Number 187560 by Mingma last updated on 18/Feb/23 Answered by mr W last updated on 19/Feb/23 $${i}\:{think}\:{there}\:{is}\:{only}\:{one}\:{possibility}. \\ $$$${blue}\:{numbers}\:{show}\:{the}\:{order}\:{in}\:{which} \\ $$$${i}\:{determined}\:{the}\:{red}\:{numbers}. \\ $$$$\mathrm{62}/\mathrm{2}=\mathrm{31}\:\:\left(\mathrm{2}\right) \\…
Question Number 121949 by liberty last updated on 12/Nov/20 $$\:\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{sum}\:\:\frac{\mathrm{2}}{\mathrm{3}+\mathrm{1}}\:+\:\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{3}^{\mathrm{2}} +\mathrm{1}}\:+\:\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{3}^{\mathrm{4}} +\mathrm{1}}+\:…+\:\frac{\mathrm{2}^{\mathrm{n}+\mathrm{1}} }{\mathrm{3}^{\mathrm{2}^{\mathrm{n}} } \:+\mathrm{1}}\:. \\ $$ Answered by bobhans last updated on…
Question Number 187390 by mathlove last updated on 18/Feb/23 $${odd}\:{number}\:\left(\mathrm{2}{n}+\mathrm{1}\right) \\ $$The sum of twelve consecutive natural numbers is 288. Find the largest natural number…
Question Number 121832 by bemath last updated on 12/Nov/20 $$\:\:{Let}\:\begin{cases}{{u}_{\mathrm{1}} =\mathrm{1}}\\{{u}_{\mathrm{2}} =\mathrm{1}}\end{cases}\:{and}\:{u}_{{n}+\mathrm{2}} \:=\:{u}_{{n}+\mathrm{1}} \:+\:{u}_{{n}} \\ $$$${find}\:{u}_{{n}} . \\ $$ Commented by bemath last updated on…
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Question Number 187235 by Rupesh123 last updated on 15/Feb/23 Commented by MJS_new last updated on 15/Feb/23 $$\mathrm{I}\:\mathrm{believe}\:\mathrm{it}'\mathrm{s}\:\mathrm{3} \\ $$ Commented by Rasheed.Sindhi last updated on…