Question Number 165777 by ZiYangLee last updated on 08/Feb/22
$$\mathrm{The}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{first}\:\mathrm{the}\:{n}\:\mathrm{terms}\:\mathrm{of}\:\mathrm{a}\:\mathrm{series} \\ $$$$\left\{{a}_{{n}} \right\}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:{S}_{{n}} =\mathrm{3}{n}^{\mathrm{2}} +{n},\:\mathrm{find}\:\underset{{k}=\mathrm{10}} {\overset{\mathrm{20}} {\sum}}{a}_{{k}} \:. \\ $$$$\left.\mathrm{A}\right)\:\mathrm{910} \\ $$$$\left.\mathrm{B}\right)\:\mathrm{913} \\ $$$$\left.\mathrm{C}\right)\:\mathrm{968} \\ $$$$\left.\mathrm{D}\right)\:\mathrm{1256}\: \\ $$
Answered by Rasheed.Sindhi last updated on 08/Feb/22
$${S}_{\mathrm{9}} =\mathrm{3}\left(\mathrm{9}\right)^{\mathrm{2}} +\mathrm{9}=\mathrm{252} \\ $$$${S}_{\mathrm{20}} =\mathrm{3}\left(\mathrm{20}\right)^{\mathrm{2}} +\mathrm{20}=\mathrm{1220} \\ $$$$\underset{{k}=\mathrm{10}} {\overset{\mathrm{20}} {\sum}}{a}_{{k}} ={S}_{\mathrm{20}} −{S}_{\mathrm{9}} =\mathrm{1220}−\mathrm{252}=\mathrm{968} \\ $$