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
Answered by JDamian last updated on 16/Feb/23
$$\mathrm{Does}\:\mathrm{it}\:\mathrm{really}\:\mathrm{exist}? \\ $$$$ \\ $$$$\mathrm{288}=\frac{\mathrm{12}}{\mathrm{2}}\left({n}+{n}−\mathrm{11}\right)\:\Rightarrow\: \\ $$$$\mathrm{48}=\mathrm{2}{n}−\mathrm{11} \\ $$$${n}=\frac{\mathrm{59}}{\mathrm{2}}\:\:\notin\:\:\mathbb{N} \\ $$$$\mathrm{19}+\mathrm{20}+\:\centerdot\centerdot\centerdot\:+\mathrm{29}+\mathrm{30}=\mathrm{294} \\ $$$$\mathrm{18}+\mathrm{19}+\:\centerdot\centerdot\centerdot\:+\mathrm{28}+\mathrm{29}=\mathrm{282} \\ $$
Commented by mathlove last updated on 16/Feb/23
$${n}=\mathrm{12}\:\:{find}\:{a}_{{n}} \\ $$
Commented by JDamian last updated on 16/Feb/23
there is no such natural number a_n
Commented by MathsFan last updated on 17/Feb/23
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$
Commented by mathlove last updated on 17/Feb/23
$$\mathrm{13}+\mathrm{15}+\mathrm{17}+\mathrm{19}+\mathrm{21}+\mathrm{23}+ \\ $$$$\mathrm{25}+\mathrm{27}+\mathrm{29}+\mathrm{31}+\mathrm{33}+\mathrm{35}=\mathrm{288} \\ $$
Commented by mr W last updated on 17/Feb/23
$${then}\:{your}\:{question}\:{is}\:{wrong}! \\ $$
Commented by JDamian last updated on 19/Feb/23
Mathlove, do you even check your questions and solutions before posting them?
Answered by HeferH last updated on 16/Feb/23
$${n}\:+\:\left({n}\:+\:\mathrm{1}\right)\:+\:\left({n}\:+\:\mathrm{2}\right)\:+\:…\:+\:\left({n}\:+\:\mathrm{11}\right)\:=\:\mathrm{288} \\ $$$$\:\mathrm{12}{n}\:+\:\mathrm{1}\:+\:\mathrm{2}+…+\mathrm{11}\:=\:\mathrm{288} \\ $$$$\:\mathrm{12}{n}\:+\:\frac{\mathrm{11}\left(\mathrm{11}\:+\:\mathrm{1}\right)}{\mathrm{2}}\:=\:\mathrm{288} \\ $$$$\:\mathrm{12}{n}\:=\:\mathrm{288}\:−\:\mathrm{66} \\ $$$$\:{n}\:=\:\frac{\mathrm{222}}{\mathrm{12}}\:=\:\frac{\mathrm{37}}{\mathrm{2}}\:\notin\:\mathbb{N}\: \\ $$
Commented by MathsFan last updated on 17/Feb/23
$$\mathrm{thanks}\:\mathrm{a}\:\mathrm{lot}\:\mathrm{sir} \\ $$