Question Number 17019 by tawa tawa last updated on 29/Jun/17
$$\underset{\mathrm{r}\:=\:\mathrm{1}} {\overset{\mathrm{3}} {\sum}}\:\mathrm{2r}\:−\:\mathrm{1}\:\:=\:\:? \\ $$
Answered by arnabpapu550@gmail.com last updated on 29/Jun/17
$$=\left(\mathrm{2}×\mathrm{1}−\mathrm{1}\right)+\left(\mathrm{2}×\mathrm{2}−\mathrm{1}\right)+\left(\mathrm{2}×\mathrm{3}−\mathrm{1}\right) \\ $$$$=\mathrm{1}+\mathrm{3}+\mathrm{5}=\mathrm{9} \\ $$
Commented by tawa tawa last updated on 29/Jun/17
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$
Commented by arnabpapu550@gmail.com last updated on 30/Jun/17
![For Σ_(r=1) ^(30) (2r−1) =1+3+5+7+∙∙∙∙∙∙ 30 terms. =((30)/2)[2×1+(30−1)×2] =30×(1+29)=900](Q17035.png)
$$\mathrm{For}\:\underset{\mathrm{r}=\mathrm{1}} {\overset{\mathrm{30}} {\sum}}\left(\mathrm{2r}−\mathrm{1}\right) \\ $$$$=\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}+\centerdot\centerdot\centerdot\centerdot\centerdot\centerdot\:\mathrm{30}\:\mathrm{terms}. \\ $$$$=\frac{\mathrm{30}}{\mathrm{2}}\left[\mathrm{2}×\mathrm{1}+\left(\mathrm{30}−\mathrm{1}\right)×\mathrm{2}\right] \\ $$$$=\mathrm{30}×\left(\mathrm{1}+\mathrm{29}\right)=\mathrm{900} \\ $$