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Question Number 94690 by abony1303 last updated on 20/May/20

Commented by abony1303 last updated on 20/May/20

Pls Help!

$${Pls}\:{Help}! \\ $$

Commented by prakash jain last updated on 20/May/20

  First term a_1 =1   a_2 , a_3 =2  a_4 ,a_5 ,a_6 =3  a_k =n for ((n(n−1))/2)<k≤((n(n+1))/2)  k=2019  n=64  ((64×63)/2)=2016<2019≤((64×65)/2)=2080

$$ \\ $$$$\mathrm{First}\:\mathrm{term}\:{a}_{\mathrm{1}} =\mathrm{1} \\ $$$$\:{a}_{\mathrm{2}} ,\:{a}_{\mathrm{3}} =\mathrm{2} \\ $$$${a}_{\mathrm{4}} ,{a}_{\mathrm{5}} ,{a}_{\mathrm{6}} =\mathrm{3} \\ $$$${a}_{{k}} ={n}\:\mathrm{for}\:\frac{{n}\left({n}−\mathrm{1}\right)}{\mathrm{2}}<{k}\leqslant\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$${k}=\mathrm{2019} \\ $$$${n}=\mathrm{64} \\ $$$$\frac{\mathrm{64}×\mathrm{63}}{\mathrm{2}}=\mathrm{2016}<\mathrm{2019}\leqslant\frac{\mathrm{64}×\mathrm{65}}{\mathrm{2}}=\mathrm{2080} \\ $$

Commented by abony1303 last updated on 20/May/20

where can I learn this kind of questions  I mean what′s the name of this theme?

$${where}\:{can}\:{I}\:{learn}\:{this}\:{kind}\:{of}\:{questions} \\ $$$${I}\:{mean}\:{what}'{s}\:{the}\:{name}\:{of}\:{this}\:{theme}? \\ $$

Commented by mr W last updated on 21/May/20

a_k =⌈(((√(8k+1))−1)/2)⌉  a_(2019) =⌈(((√(8×2019+1))−1)/2)⌉=64

$${a}_{{k}} =\lceil\frac{\sqrt{\mathrm{8}{k}+\mathrm{1}}−\mathrm{1}}{\mathrm{2}}\rceil \\ $$$${a}_{\mathrm{2019}} =\lceil\frac{\sqrt{\mathrm{8}×\mathrm{2019}+\mathrm{1}}−\mathrm{1}}{\mathrm{2}}\rceil=\mathrm{64} \\ $$

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