Question Number 21504 by mondodotto@gmail.com last updated on 25/Sep/17
Answered by $@ty@m last updated on 26/Sep/17
$${ATQ} \\ $$$$\boldsymbol{{S}}_{{n}} =\left(−\mathrm{1}\right)+\mathrm{1}+\mathrm{3}+\mathrm{5}+……+\left(\mathrm{2}{n}−\mathrm{3}\right) \\ $$$$\left({i}\right)\:\boldsymbol{{S}}_{\mathrm{50}} =\frac{\mathrm{50}}{\mathrm{2}}\left\{\left(−\mathrm{1}\right)+\left(\mathrm{2}×\mathrm{50}−\mathrm{3}\right)\right\} \\ $$$$=\mathrm{25}\left(−\mathrm{1}+\mathrm{97}\right) \\ $$$$=\mathrm{25}×\mathrm{96} \\ $$$$=\mathrm{2400} \\ $$$$\left({ii}\right)\:\boldsymbol{{S}}_{{n}} =\mathrm{624} \\ $$$$\frac{{n}}{\mathrm{2}}\left\{\mathrm{2}{a}+\left({n}−\mathrm{1}\right){d}\right\}=\mathrm{624} \\ $$$$\frac{{n}}{\mathrm{2}}\left\{−\mathrm{2}+\left({n}−\mathrm{1}\right)×\mathrm{2}\right\}=\mathrm{624} \\ $$$${n}\left(−\mathrm{1}+{n}−\mathrm{1}\right)=\mathrm{624} \\ $$$${n}\left({n}−\mathrm{2}\right)=\mathrm{624} \\ $$$${n}^{\mathrm{2}} −\mathrm{2}{n}−\mathrm{624}=\mathrm{0} \\ $$$${n}=\frac{\mathrm{2}\pm\sqrt{\mathrm{4}+\mathrm{2496}}}{\mathrm{2}} \\ $$$${n}=\frac{\mathrm{2}\pm\mathrm{50}}{\mathrm{2}} \\ $$$${n}=\mathrm{26},−\mathrm{24} \\ $$$${n}=\mathrm{26}\:\left\{\because{n}\in\mathrm{N}\right\} \\ $$