Question Number 94708 by abony1303 last updated on 20/May/20
Commented by abony1303 last updated on 20/May/20
$${I}'{m}\:{so}\:{tired}\:{seeing}\:{these}\:{type}\:{of}\:{problems} \\ $$$${in}\:{my}\:{exam}.\:{Can}\:{smn}\:{explain}\:{these}\:{kind} \\ $$$${of}\:{questions}\:{in}\:{easy}\:{way}?\:{Or}\:{suggest}\:{me} \\ $$$${the}\:{book}\:{to}\:{get}\:{information}.\:{pls}. \\ $$
Answered by Worm_Tail last updated on 14/Jul/20
$${S}_{{n}} =\mathrm{2}+\mathrm{2}+\mathrm{4}+\mathrm{4}+\mathrm{4}+\mathrm{4}+\mathrm{6}+\mathrm{6}+\mathrm{6}+\mathrm{6}+\mathrm{6}+\mathrm{6}+...+{n}+{n}+{n} \\ $$$${S}_{{n}} =\mathrm{2}\left(\mathrm{2}\right)+\mathrm{4}\left(\mathrm{4}\right)+\mathrm{6}\left(\mathrm{6}\right)+...+\mathrm{2}{n}\left(\mathrm{2}{n}\right) \\ $$$${S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{2}{r}\left(\mathrm{2}{r}\right)\Rightarrow \\ $$$${S}_{{n}} =\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\mathrm{4}{r}^{\mathrm{2}} \:\: \\ $$$${S}_{{n}} =\mathrm{4}\Sigma{r}^{\mathrm{2}} \Rightarrow{S}_{{n}} =\mathrm{4}\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$$$=\frac{\mathrm{2}\left(\mathrm{2019}\right)\left(\mathrm{2020}\right)\left(\mathrm{4039}\right)}{\mathrm{3}}=\frac{\mathrm{2}\left(\mathrm{2019}\right)\left(\mathrm{4039}\right)}{\mathrm{3}}\left(\mathrm{2020}\right) \\ $$$$\frac{{S}_{\mathrm{2019}} }{\mathrm{10}}=\frac{\mathrm{2}\left(\mathrm{2019}\right)\left(\mathrm{4039}\right)}{\mathrm{3}}\left(\mathrm{202}\right)\:\:{remainder}\:\:\mathrm{0} \\ $$$$ \\ $$