Question Number 31771 by rahul 19 last updated on 14/Mar/18
$${Consider}\:{a}\:{sequence}\:{in}\:{the}\:{form}\:{of} \\ $$$${groups}\:\left(\mathrm{1}\right),\left(\mathrm{2},\mathrm{2}\right),\left(\mathrm{3},\mathrm{3},\mathrm{3}\right),\left(\mathrm{4},\mathrm{4},\mathrm{4},\mathrm{4}\right), \\ $$$$\left(\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5}\right),............ \\ $$$${then}\:{the}\:\mathrm{2000}{th}\:{term}\:{of}\:{the}\:{above}\: \\ $$$${sequence}\:{is}\::\:? \\ $$
Commented by Joel578 last updated on 14/Mar/18
$$\mathrm{1},\mathrm{2},\mathrm{2},\mathrm{3},\mathrm{3},\mathrm{3},\mathrm{4},\mathrm{4},\mathrm{4},\mathrm{4},\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5},\mathrm{5},... \\ $$
Answered by Tinkutara last updated on 14/Mar/18
$${Let}\:{S}_{{n}} =\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\left({n}\:{terms}\right) \\ $$$${S}_{{n}} =\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$${Now}\:{observe}\:{S}_{\mathrm{4}} =\mathrm{10}.\:{It}\:{means}\:\mathrm{10}^{{th}} \:{term} \\ $$$${is}\:{the}\:{last}\:``\mathrm{4}''. \\ $$$${Similarly}\:{S}_{\mathrm{5}} =\mathrm{15}\:{means}\:\mathrm{15}^{{th}} \:{term} \\ $$$${is}\:{the}\:{last}\:``\mathrm{5}''. \\ $$$$\vdots \\ $$$$\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}=\mathrm{2000} \\ $$$${n}\approx\mathrm{62}.\mathrm{75} \\ $$$${S}_{\mathrm{62}} =\mathrm{1953}\Rightarrow\mathrm{1953}^{{rd}} \:{term}\:{is}\:{the}\:{last}\:\mathrm{62}. \\ $$$${S}_{\mathrm{63}} =\mathrm{2016}\Rightarrow\mathrm{2016}^{{th}} \:{term}\:{is}\:{the}\:{last}\:\mathrm{63}. \\ $$$${In}\:{between},\:\mathrm{2000}^{{th}} \:{term}\:{is}\:\mathrm{63}. \\ $$
Commented by rahul 19 last updated on 14/Mar/18
$$\boldsymbol{\mathcal{W}}\mathbb{O}\mathscr{W}\:\:! \\ $$